Accelerator is the strongest Esper in Academy City. While he's far from being the strongest character in "To Aru/A Certain" franchise, he's still ranked #1 Level 5, (which is the current strongest level which only 7 Espers have reached.) He's also the only Esper that could reach Level 6 in theory, and his presence plays a major role in the development of the setting.
The "Accelerator" is actually the name of his ability but everyone referred to him as such for a long time that he forgot his original name.
Accelerator's ability is one of the most over-powered and unique abilites in the existence: Vector Manipluation! He has the power to control any vector that comes in contact with him. That's the only thing he can do, but it also means he can do anything.
A Vector is any quantity that has both Magnitude and Direction.
As long as something has a Magnitude and a Direction, Accelerator can manipulate it: Not only can direct it anywhere he pleases, he can also change its size/power in any way he pleases. Since everything in the physical world has a vector quantity in it, Accelerator can manipluate almost anything as long as it touches him.
So, he can relflect atttacks or redirect them. He can return projectiles stronger to the opponent! He can create plasma in air! He can (and does) reverse the blood flow of his victims. He can even manipulate sound and light waves because they contain vectors too.
Accelerators powers can also work subconsiusly and even in his sleep. By default, there's a barrier around him that reflects any attack!
Limitations and Weakness
Despite being able to manipulate all vectors, there are limits to what he can do. First, he has to calculate all the vectors mathmatics in his head, in real time! That means if there are too many vectors for him to calculate he won't be able to do that, but his brain is so overpowered too! Accelerator never reached this limit until the time he got shot and got brain injury that made him rely on a computer.
What makes Accelerator so powerful isn't that his powers are limitless, it's that the limits are so high for him that they don't matter. After his brain injury, he can now only use his powers for less than hour a day, and even when he use them, he lost half of his calculating ability. The super-computer assisting his brain can't calculate half as fast as his brain used to calculate before the injury. (Let that sink in your head.) That's the limit the author put for him, otherwise he's basically invincible!
One of the downsides of Accelerator powers is that he has to allow unharmful things like sound and air to bypass his barrier. It's stated that his white hair resulted from subcounciously reflecting the sun's UV rays. This also means that Accelerator can't be affected by new substances even if they're positive for him unless he reprogrammed his barrier to let them pass.
Another weakness for him is that he can only manipulate vectors! Despite seeming invincible, Accelerator's powers are still only capable of doing that one thing. If he was hit with an attack that has a vector element, his power can do nothing against it.
For example, let's say some Magician directly attacks Accelerator internals, that will instantly kill him.
Of course, there's also the brain injury!
Early in the series, Accelerator got shot while he couldn't activate his powers. His brain injured so hard that he couldn't speak anymore without a computer assisting his brain. The device's battery can only less than an hour in super-power mode. This limitation results in a highly tactical situations where Accelerator needs to fight without his powers, reserving battery before fighting stronger foes.
Accelerator's final weakness is his body. He's so used to his power that once he's stripped out of them, he becomes weaker than a normal human. He gets better later in the story because he learned to not rely on his powers and that's why he got fire-arms training.