MTHFR!

You may be reading this because you've already seen the topic brought up in a natural living group or mommy group somewhere. Maybe your doctor told you about it. Maybe it's 2 am and you're just browsing random stuff on the internet, either way, it doesn't really matter because you're here now. MTHFR is an acronym (short name) for; methylenetetrahydrofolate reductase. Fun right? Don't worry I can't pronounce either. I actually like to refer to it as the motherf*cker gene because your life can potentially be made more complicated by it. This gene determines your ability to run a specific set of amino acid conversions, these conversions are pretty important for overall health, particularly given the man-made chemical soup we live in today. The primary function of this gene is to convert folate from a passive form into an active form in the body. Without this, the negative effects can cascade in the body, from short term effects like how prescriptions affect us, to long term impacts such as the increased risk for schizophrenia.
The next question should be why is folate important? Without proper folate levels in the body, we can't properly make RNA, DNA, or metabolize amino acids (proteins). This effects healing from injury or illness, glutathione production (we will get into this a bit later), and normal cell division.
Example: Chronic inflammation is associated with being the root cause of multiple chronic illnesses such as heart disease. If the body cannot utilize folate due to an MTHFR gene mutation then the constant repair required to keep the body running is not going to happen at a proper rate. AKA you end up with a heart attack or stroke because of constant damage to blood vessels that only get patch jobs with cholesterol. Cholesterol is only meant to be a temporary patch in the blood vessels, not a permanent solution. Obviously, chronic inflammation can come from many sources, this is only a simple example of how the slowed down repairs can have a dramatic effect.
Example 2: In the brain, you have normal pruning of unused neural pathways but without the normal rate of regeneration, new neural pathways are not created at the same rate (Alzheimer's or dementia). Actually, the effects in the brain overall can be very dramatic, I've linked an article at the end focusing on just this.
Here is an example of the impact folate can have on children.

Folinic acid improves verbal communication in children with autism and language impairment: a randomized double-blind placebo-controlled trial
"We sought to determine whether high-dose folinic acid improves verbal communication in children with non-syndromic autism spectrum disorder (ASD) and language impairment in a double-blind placebo control setting. Forty-eight children (mean age 7 years 4 months; 82% male) with ASD and language impairment were randomized to receive 12 weeks of high-dose folinic acid (2 mg kg−1 per day, maximum 50 mg per day; n=23) or placebo (n=25). Children were subtyped by glutathione and folate receptor-α autoantibody (FRAA) status. Improvement in verbal communication, as measured by a ability-appropriate standardized instrument, was significantly greater in participants receiving folinic acid as compared with those receiving placebo, resulting in an effect of 5.7 (1.0,10.4) standardized points with a medium-to-large effect size (Cohen’s d=0.70). FRAA status was predictive of response to treatment. For FRAA-positive participants, improvement in verbal communication was significantly greater in those receiving folinic acid as compared with those receiving placebo, resulting in an effect of 7.3 (1.4,13.2) standardized points with a large effect size (Cohen’s d=0.91), indicating that folinic acid treatment may be more efficacious in children with ASD who are FRAA positive. Improvements in subscales of the Vineland Adaptive Behavior Scale, the Aberrant Behavior Checklist, the Autism Symptom Questionnaire and the Behavioral Assessment System for Children were significantly greater in the folinic acid group as compared with the placebo group. There was no significant difference in adverse effects between treatment groups. Thus, in this small trial of children with non-syndromic ASD and language impairment, treatment with high-dose folinic acid for 12 weeks resulted in improvement in verbal communication as compared with placebo, particularly in those participants who were positive for FRAAs."

Translation:
The researchers wanted to see if folinic acid (folate) could improve speech in children with speech delay, mostly in children with autism. There was a total of 48 children, mostly boys. They were placed into 2 groups, 1 received highly daily doses of folate and the other got a placebo. They also tracked a couple of gene markers for glutathione and folate in the children. The trial lasted for 12 weeks and the children who received folate showed significant improvement. The children who also had the gene markers showed the biggest improvement. There were no serious adverse events in either test group.

Let's dig into the glutathione aspect a bit. First, let's define glutathione. Wikipedia did such a good definition I'm just going to quote them:
"Glutathione (GSH) is an important antioxidant in plants, animals, fungi, and some bacteria and archaea. Glutathione is capable of preventing damage to important cellular components caused by reactive oxygen species such as free radicals, peroxides, lipid peroxides, and heavy metals."

Let's unpack that a bit shall we? Glutathione is needed to prevent damage to parts of the cell. The damage can come from heavy metals such as aluminium, lead, or mercury. The damage can also come from free radicals, oxygen-containing molecules (peroxide), and the fat-containing oxygen-containing molecules. There may be other sources of damage not listed.
It helps prevent the damage by breaking down specific chemical bonds in the offending substances, thereby rendering them inert and more easily removed from the body.
So what does MTHFR have to do with glutathione? Well, if you have a mutation on this gene it may affect the total production of glutathione. Our bodies require specific amino acids to synthesize glutathione if our ability to absorb these amino acids is reduced then our capacity to make glutathione is also reduced. These highly specific processes are laid out in the Wikipedia article linked above.

Next, let's focus on the amino acid aspect a bit more. Amino acids are the building blocks of the proteins our bodies use to make collagen, muscle, and many other things in our bodies. The human body can make roughly 20 of the amino acids we need but the remainder must found in our diet. Remember at the beginning when we read that it affects our ability to absorb them? A certain amount does get absorbed otherwise we wouldn't live to become adults, it's just happening at a reduced rate. The implications of a reduced absorption rate are far-reaching and most easily seen in the elderly. Older people who are not getting enough essential amino acids can suffer cognitively and in their overall physical health, in short, the decline of age will be faster and more dramatic. The lowered absorption rate is exacerbated by how our food is raised/grown and then processed, which lowers the overall nutritional quality, thereby giving us less to work with, in the first place.
Note: this also affects children and non-elderly adults, it's just easier to see in the elderly.

The last thing to cover before we get into the actual mutations is clarifying why the ability to make RNA and DNA is relevant beyond making babies. Whenever we have damage to the body in any form we will need to discard the damaged cells and then build new cells to replace them. We also discard aged cells which are also replaced with new ones. In both of these scenarios, the new cells require DNA in them so they know what to do and when their function has been served. Every new cell that our body makes contains a copy of our DNA, which means we are constantly making new copies. As for RNA, this is predominantly viewed as being the messenger/delivery service that our DNA uses to convey information from its code into actual functions.
When it comes to specific gene mutations there are currently 50 recognized variations on the MTHFR gene, however, 2 specific mutations are studied more heavily than others. When we inherit our DNA we get half from each parent so I just want to clarify some vocabulary you may run into if you decide to keep reading beyond this post.

Heterozygous = one copy of the gene. Homozygous = two copies.
First, we will look at the C677T mutation. This particular mutation has actual numbers showing how much function of the gene is lost, it ranges between 40% - 70% depending on whether or not you have 1 or 2 copies (source below).
It is unclear if people with the A1298C mutation regularly experience loss of gene function based on current research. However, if someone is compound heterozygous (one copy of each) there is an estimated 50% loss of function. These numbers are based on Dr. Ben Lynch and his research on MTHFR gene mutations.
It is important to note that if you choose to supplement folate that you must be sure to get a whole food type and not a synthetic isolate. This is because folic acid will still take up that same molecular location but not be usable by the body, taking up the same spot means it will also block absorption of the useful form of the vitamin.
The only way to know if you have a mutation is to get a DNA test. Your doctor can do this however getting them to order the test might be challenging. A commercial DNA test (linked below) can give you the same info, but you do need to know that they will have some legal right to the data.
I've kept the writing fairly trim without any commentary along the way, this is to simplify the learning process of a very technical and detailed topic. I hope this is helpful in your research process.

Resources used in writing this piece:
https://ghr.nlm.nih.gov/gene/MTHFR#
https://bit.ly/2rfjyom
https://www.omim.org/entry/607093
https://www.ncbi.nlm.nih.gov/books/NBK6561/
https://www.nature.com/articles/mp2016168
http://www.aminoacid-studies.com/amino-acids/what-are-amino-acids.html
http://genetics.thetech.org/ask/ask24
https://www.nature.com/scitable/topicpage/rna-functions-352
http://mthfr.net/mthfr-research/2012/01/27/