(My answer is very long so I hope not to bore you interested readers)
Motivation
In the not too distant future, population growth, global warming, and the need for natural resources will possibly create pressure on humans to consider the colonization of new habitats. Although the surface of the Earth's oceans, the marine depths, are possible options, the Earth's orbital space close to the planet, the moon and nearby planets is also considered, as well as the creation of mines in the solar system in order to extract energy and materials.
Through terraforming, humans could turn the planet Mars into habitable long before there was extreme need. Mars is at the limit of the habitable zone, so it could give humanity a few thousand additional years to develop a superior space technology, to be able to settle on the edges of the solar system and other systems.
Background
It is believed that once Mars had an environment relatively similar to that of the Earth at the beginning of its history, with a dense atmosphere and abundant water that was lost over millions of years; It has even been suggested that this process could be cyclical.
The similarity is given by the thickness of the Martian atmosphere, as well as the evident presence of water in a liquid state on the planet at some point in its past. The atmosphere, after millions of years, has decreased due to the escape of gases into space, although it has also partially condensed in solid form. Although it appears that water existed on the Martian surface, it is now only found at the poles and just below the surface of the planet in the form of permafrost.
The exact mechanism of this loss is not yet clear, although many theories have been proposed. The lack of a magnetosphere surrounding Mars may have allowed the solar wind to erode the atmosphere, the relative low gravity of Mars would help accelerate the loss of light gases in space. The evident lack of tectonic plates is another quite plausible factor, since a lack of tectonic activity, in theory, would make the recycling of the gases trapped in the sediments of the soil reverting them to the atmosphere much slower. The absence of a magnetic field and geological activity may be the result of the smaller size of Mars, allowing its interior to cool faster than Earth, although the details of such processes are still unknown. However, none of these processes is likely to be significant throughout the life of most animal species, or even in the time scale of human civilization, and the slow loss of the atmosphere may be counteracted through artificial terraforming activities.
Mars, in and of itself, already contains many of the minerals that could theoretically be used for terraforming. Additionally, recent investigations have discovered large amounts of ice in the form of permafrost just below the Martian surface to latitude 60, as well as on the surface of the poles, where it is mixed with dry ice and frozen CO2. It has also been hypothesized that there are large amounts of ice in the lower layers of its surface. As the Martian summer arrives, the frozen carbon dioxide (CO2) from the poles returns to the atmosphere, and the small amount of wastewater is swept from there by winds approaching 250 miles per hour (402 km/h). This seasonal event transports large amounts of dust and water vapor into the atmosphere, giving rise to cirrus clouds very similar to terrestrial clouds.
Oxygen is only present in the atmosphere in minimal amounts, but it is present in large quantities in metal oxides on the Martian surface. There is also some oxygen present in the soil in the form of nitrates.
The analysis of soil samples obtained by the Phoenix Lander indicated the presence of perchlorate, which is used to release oxygen in chemical oxygen generators. Additionally, electrolysis could be used to separate the water of the planet in oxygen and hydrogen if sufficient electricity existed.
Theoretical methods of terraforming
The terraforming of Mars would involve two intertwined changes: creation of an atmosphere and keeping the planet warm. The Martian atmosphere is relatively thin, which makes the pressure at the surface very low (0.6 kPa), compared to that on Earth (101.3 kPa). The atmosphere of Mars consists of 95% carbon dioxide (CO2), 3% nitrogen, 1.6% argon, and only contains small amounts of oxygen, water, and methane. Because its atmosphere is mainly made up of CO2, a known gas that produces the greenhouse effect, once the planet began to heat up and melt the reserves of the poles, a greater amount of CO2 would enter the atmosphere causing this greenhouse effect to increase . Each of the two processes would favor the other, helping, in this way, terraforming. However, it would be necessary to apply certain techniques in a controlled and large-scale way for a long enough time to achieve sustainable changes and to turn this theory into reality.
Reconstruction of the atmosphere
Since ammonia is a potent greenhouse gas, and it is possible that large amounts of this frozen compound have naturally accumulated in objects the size of asteroids orbiting the outer solar system, it would be imaginable to move them and send them into the atmosphere of Mars. The collision of a comet on the surface of the planet would cause a destruction that could be counterproductive. Instead, by aerobraking, if possible, it would be possible for the frozen mass of the comet to vaporize and become part of the atmosphere it traverses. A bombardment of small asteroids would increase both the mass of the planet and its temperature and atmosphere.
The need for an inert gas is a challenge that the atmosphere builders will have to address. On Earth, nitrogen is the main atmospheric component, constituting 79% of it. Mars would require a similar inert gas, although not necessarily in so much quantity. In any case, obtaining significant amounts of nitrogen, argon or other non-volatile gases could be complicated.
The import of hydrogen could be carried out by atmospheric and hydrospheric engineering. Depending on the level of carbon dioxide in the atmosphere, the import and reaction with hydrogen would produce heat, water and graphite through the Bosch reaction. Adding water and heat to the environment would be the key to making the dry and cold world suitable for terrestrial life. Alternatively, reacting hydrogen with carbon dioxide by the Sabatier reaction would produce methane and water. The methane could be released into the atmosphere where it would complement the greenhouse effect. Presumably, hydrogen could be obtained, in quantities, from gaseous giants or extract it from objects in the outer solar system that have hydrogen-rich compounds, although the amount of energy needed to transport the necessary amount would be large.
Create an atmosphere with water
The most important way to create an atmosphere on Mars is by importing water. Obtaining it from the ice of the asteroids, or the moons of Jupiter or Saturn. Adding water and heat to the Martian environment is a vital point in making this cold and dry planet suitable for sustaining life.
Water sources
An important source of water nearby is the dwarf planet Ceres, which, according to studies, occupies between 25 and 33% of the asteroid belt.
The mass of Ceres is approximately 9.43 x 1020kg. Estimates about the amount of water that this planet may have vary considerably, but 20% is a typical amount among those estimated. In addition, it is thought that a large amount of this water is at the surface or near surface level of the planetoid. Using the estimates that we have just given, the water mass of Ceres is approximately 1,886 x 1020kg. The total mass of Mars is approximately 6.4185 x 1023kg.
Therefore, and making estimated calculations, the water that could be in Ceres would be equivalent to 0.03% of the total mass of Mars.
The transport of a significant amount of this water, or water in general from any of the icy moons, would be quite a challenge. On the other hand, any attempt to disturb the Ceres orbit to add the planetoid to the planet Mars (similar to the strategy of using gravitational traction to divert the asteroids), thus increasing the Martian mass a tiny fraction, but at the same time adding a significant amount of heat (since Ceres is not a small celestial body), could cause a disturbance in the Martian orbit in addition to prolonged geological changes, such as the restoration of hydrostatic balance, caused even by the softest of impacts.
Import of ammonia
Another method, much more complicated, would be to use ammonia as a powerful greenhouse gas (since it is possible that nature has large reserves of it frozen in asteroids orbiting the outskirts of the solar system); it might be possible to move these asteroids (for example by using large nuclear bombs to exploit them and make them move in the right direction) and send them into the Martian atmosphere. Since ammonia (NH3) has a lot of nitrogen, perhaps it could solve the problem of having a gas buffer in the atmosphere. Repeated small impacts could also help to increase the temperature and mass of the atmosphere.
The need for a buffer gas is a challenge that will face all potential atmospheric builders. On Earth, nitrogen is the primary atmospheric component, constituting up to 79% of it. Mars would require a similar component of gas buffer, although not necessarily in such a high quantity. Even so, obtaining significant amounts of nitrogen, argon or some other comparatively inert gas would be quite complicated.
Import of hydrocarbons
Another way would be to import methane or other hydrocarbons, (which are common in Titan's atmosphere and on its surface). The methane could be vented to the atmosphere where it would act as a component of the greenhouse effect.
Methane (and other hydrocarbons) can also be useful to produce a rapid increase in the pressure of the insufficient Martian atmosphere. In addition, these gases can be used for the production (in the next step of the terraforming of Mars) of water and CO2 for the Martian atmosphere, by the reaction: CH4 + 4 Fe2O3 => CO2 + 2 H2O + 8 FeO
This reaction could probably be initiated by heat or by Martian UV solar irradiation. Large quantities of the resulting products (CO2 and water) are necessary to initiate photosynthetic processes.
Import of hydrogen
The importation of hydrogen can also be done for the engineering of the atmosphere and the Hydrosphere. For example, hydrogen could react with iron (III) oxide on the Martian surface, which would give water as a product: H2 + Fe2O3 => H2O + FeO
Depending on the level of carbon dioxide in the atmosphere, the import and reaction of hydrogen produces heat, water and graphite through the Bosch reaction. Alternatively, hydrogen reacts with the carbon dioxide atmosphere through the Sabatier reaction to produce methane and water.
Use of fluorine compounds
Because long-term climate stability would be required to sustain a human population, the use of especially potent fluorine greenhouse gases has been suggested, possibly including sulfur hexafluoride or halocarbons such as chlorofluorocarbons (or CFCs) and perfluorocarbons (or PFC).
These gases are proposed for introduction because they produce a greenhouse effect many times stronger than CO2. This can possibly be done by sending rockets with payloads of compressed CFCs in collision courses with Mars. When the rockets hit the surface, they would release their payloads into the atmosphere. A constant bombardment of these "CFC rockets" would have to be sustained for a little over a decade while Mars changes chemically and becomes warmer. However, its useful life due to photolysis would require an annual replacement of 170 kilotons, and would destroy any ozone layer.
To sublimate the CO2 glaciers of the polar south, Mars would require the introduction of approximately 0.3 microbars of CFCs into the Martian atmosphere. This equates to a mass of approximately 39 million metric tons. This is roughly three times the amount of CFCs manufactured on Earth from 1972 to 1992 (when CFC production was banned by an international treaty). The mineralogical prospects for Mars estimate the elemental presence of fluorine in the massive composition of Mars at 32 ppm in mass compared to 19.4 ppm for Earth.
A proposal to extract fluorine-containing minerals as a source of CFCs and PFCs is supported by the belief that because these minerals are expected to be at least as common on Mars as they are on Earth, this process could sustain the production of sufficient amounts of optimal greenhouse gasses (CF3SCF3, CF3OCF2OCF3, CF3SCF2SCF3, CF3OCF2NFCF3, C12F27N) to keep Mars at "comfortable" temperatures, as a method to maintain an Earth-like atmosphere previously produced by other means.
Elevation of temperature
Mirrors made of extremely fine aluminized mylar could be placed in orbit around Mars to increase the total insolation it receives. This would increase the temperature directly, and also vaporize water and carbon dioxide to increase the greenhouse effect on the planet.
Although generating halocarbons on Mars could contribute to adding mass to the atmosphere, the main function would be to capture incident solar radiation. Halocarbons (such as CFCs and PFCs) are potent greenhouse gases, and are stable in the atmosphere for prolonged periods of time. They could be produced by genetically modified aerobic bacteria or by mechanical devices spread over the surface of the planet.
Modifying the albedo of the Martian surface would also be a way to take advantage of incident light more efficiently. Altering the color of the surface with a dark dust like soot, dark microbial life forms or lichens would serve to transfer a large amount of solar radiation to the surface in the form of heat before it was reflected back into space. The use of life forms is particularly attractive since they could spread themselves.
Nuclear bombing
Nuclear bombardment of the crust and polar ice caps has been suggested as a quick and dirty method of warming the planet.
If a nuclear device were detonated in the polar regions, the intense heat would melt large quantities of frozen water and carbon dioxide.
The gases produced would densify the atmosphere and contribute to the greenhouse effect. Additionally, the dust raised by the nuclear explosion would cover the ice and reduce its albedo, allowing it to melt more quickly under the sun's rays. The detonation of a nuclear device beneath the surface would heat the crust and help degas the carbon dioxide trapped in the rocks. Although nuclear bombs are attractive in the sense that they make use of dangerous and obsolete weapons on Earth and add heat to the planet quickly and economically, it carries the negative connotations of mass destruction to the native environment and potential pernicious effects of nuclear disintegration.
Magnetic shield between Mars and the Sun
During the 2050 Planetary Science Workshop at the end of February 2017, NASA scientist Jim Green proposed to launch a magnetic shield between the planet and the Sun to protect it from high-energy solar particles, it will be located at a point of Lagrange L1 (a relatively stable point between Mars and the Sun) approximately 320 R (radius of Mars). The shield would allow the planet to restore its atmosphere, quickly improving its habitability. The simulations indicate the planet would be able to achieve half the atmospheric pressure of the Earth in a matter of years, and not centuries or millennia. Without the solar winds affecting the planet, the carbon dioxide frozen in the ice sheets of each pole would begin to sublimate and heat the equator. The layers of ice would begin to melt to form an ocean.