To elaborate on some of what has been said..and at the risk of TL;DR...and with the qualification that I am not a virologist, I offer my opinion also.
The virus is not alive. It is a pathogen that enters a normal cell and hijacks it to produce copies of itself. The infected cell is no longer doing what it is supposed to be doing and if you have enough infected cells, you have problems.
The more copies you make the more likely that you will get a copy error. That is, the copy is not identical to the virus that infected the cell to make copies. That is what the mutation is; an error in the copy. Those "errors" can then go on to engineer reproduction of more of the "erroneous" copies.
Only a portion of the mutations are “biologically significant”. Minor mutations are sometimes used forensically to monitor pandemic spread and the like. Some mutations could significantly decrease the ability to infect and, thus, decrease spread. Mutations could, potentially, make a virus that is much more deadly. Those could, paradoxically, also decrease spread because the host does not stay alive long enough to easily spread the infection (something like Ebola, in comparison to SARS-CoV-2).
Some mutations could make the virus more transmittable and that’s what the concern is here. Simplistically and somewhat figuratively, the mutation could produce longer and sharper spikes; functionally enhancing penetration into the cell.
The mRNA vaccines are unique in that they carry instructions to cells to make a compound. Literally, they are mRNA (instructions) attached to lipids (to get the cell to take them in) where they instruct the cell to make and release this compound. It’s a program, written in genetic C if your cells were a C compiler with a CPU– crazy cool stuff.
The presence of that compound is designed to stimulate your immune system (which is multidimensional and much more than just T-cells) to look for and neutralize that compound. Within important limits, anything else that looks like that compound also gets destroyed. Therein lies the vaccination; create an immune response that will include recognition of the virus. So, although, the vaccine does not target anything directly, the compound it instructs the cell to make has some characteristics in common with the virus – like a sequence of proteins that are in common with a subset of the virus’ spike proteins.
At this point, I am unaware of ANY evidence that mutations that have spread are resistant to the immune activity created by the mRNA vaccines. But, this can be evaluated and is being evaluated because everyone expected mutations. Pfizer has already released some early data
https://www.biorxiv.org/content/10.1101/2021.01.07.425740v1 showing that their vaccine is effective against one mutation. That is what the vaccine makers will be doing; testing their efficacy against mutations.
It is like when somebody you know wears an eye patch. You still know who it is even though some of the face is obscured. Wear a full ski mask and maybe then you don’t recognize them.
We may get to a point, at some time, when the mRNA program has to be rewritten, and that is likely what they would do, probably including a mixture; a “multi strain” vaccination like we see with the seasonal flu.
Edit: The NYT just published an in depth and illustrated article about the mutations if you are interested. Very well done I think.
https://www.nytimes.com/interactive/2021/health/coronavirus-mutations-B117-variant.html