Some parts of this post are moderately speculative and based on a general understanding of evolution and not a deep understanding of viruses. Take with a pinch of salt, and I welcome criticism of these thoughts.
I will refer to three mutations: N501Y, E484K and K417N.
And to three variants of the virus: the UK variant (B.1.1.7), the South Africa variant (B.1351) and the Brazil variant (P.1). I'll skip the dots and say B117, for example.
1) Mutations happen spontaneously when nucleic acids like DNA and RNA are copied, because the process is imperfect and some of those copies have errors (to oversimplify a bit). Most mutations tend to be very bad for the organism (I mean the virus) because they stop some part of it from working and many parts are necessary. These mutations are immediately 'lost' because the organism (the virus) cannot reproduce itself, so we never get to see these mutations that are damaging to the virus even though they are the majority. A small number of mutations will tend to be only slightly bad; they will arise and then slowly die out over a few generations because they are less good than their relatives without the mutation. Some mutations have no real effect and just get passed on without changing anything. A very tiny proportion can make the organism better (for the virus that is, it would be worse for us).
2) This is a bit more complicated with a new virus like this one. An organism that has been around for a long time has evolved and adapted to its environment. Evolution never stops, but the easy adaptations - the simple changes that cause big improvements - have already happened, and so it's extremely rare for a mutation to help much. But a new virus has not had the time to do that. It's suddenly experiencing a totally new and different environment after many generations of adapting to life in bats. So there are many more possible ways for it to get better at growing and spreading in humans (that is, make it worse for us), and the number of possible mutations that can help it get better is likely to be much larger. We are very lucky that this virus mutates quite slowly. We've had hundreds of millions of infections so far and perhaps trillions of virus particles generated, but only a tiny handful of mutations have arisen that have clearly helped the virus i.e. made it worse for us.
3) We cannot rely on evolution to make this virus safer for us. A common but incorrect idea is that the virus will necessarily evolve to become less deadly over time. There is some evolutionary logic to this notion. A dead host cannot transmit the virus, the thinking goes, so variants that are less likely to kill their hosts should be more successful at being transmitted. These variants should therefore take over from more lethal ones. It's a nice story but there's little evidence that this happens in other viral diseases. And even the logic fails completely with Covid. The reason is that people infected with Covid pass on the virus mostly within the first week of their infection. If they die, it happens about 2-3 weeks after that. So whether the infected person lives or dies does not affect transmission of the virus. Evolution will not necessarily push the virus to become less lethal because it does not even 'see' whether the infection was lethal or not. It might become less lethal. But the opposite is perhaps equally possible: the virus could evolve to become more lethal. There's a small amount of evidence from the UK that this has happened. People infected with the UK variant there are thought to have a death rate that was 30% higher than previous strains (more evidence needed to be confident in this). I should note that a possible 30% increase in the probability of dying from Covid is very bad news, but it's not as terrifying as it sounds at an individual level. It's about how much worse your risk of dying from Covid would be if you were ~2 to 3 years older.
4) Of the many mutations we have seen in the viruses we have sequenced from around the world, the most noteworthy right now are mutations N501Y, E484K and K417N. You'll probably hear more about them in the next few weeks. The names denote the position and nature of the change in a protein. For example, N501Y means that a mutation changed the amino acid at the 501st position on a protein from asparagine (symbol N) to tyrosine (symbol Y). All three of these mutations alter a specific part of the 'spike' protein that the virus uses to attach itself to our cells. Some of our vaccines are designed specifically to elicit antibodies from us that will attack this protein in the virus. This is because it's a distinctive target (different from other viruses and antigens), the virus needs it to have a particular shape to attach to our cells, and because of this it probably evolves more slowly than other parts of the virus. But this protein has evolved in several virus variants, and the changes have had a variety of effects.
5) To simplify some terminology: a variant / lineage / strain is a group of viruses that have descended from a common ancestor and share similar sets of mutations. If new mutations arise in a variant that make it seem different enough, then we deem the viruses with those new mutations a new variant. We give the variant a new name, just like we name strains of wheat or rice. But the way we name virus variants is dizzyingly complicated and unintuitive. There are multiple scientific names, and some of them are named after important mutations (which can occur in multiple variants, remember). So the 'UK variant' is also called B117 by some scientists and 501Y.V1 by others. Geographical names are considered bad because they can lead to negative publicity for the associated place. But the alphanumeric gibberish of B117, B1351 and B11248 (renamed P1) is incomprehensible at a time when we desperately need clarity. In the future I think we should name these like hurricanes or after random objects (apple/banana/cantaloupe). For now I would stick with the geographical names for simplicity.
6) From some fantastic lab studies, we have a pretty good understanding how some mutations have caused a few variants to spread more quickly. The mutation N501Y evolved independently in all three variants we are most alarmed about - the UK variant B117, the South Africa variant B1351, and the Brazil variant P1. Remember that mutations arise at random, so the fact that this one arose independently in 3 variants that all spread more rapidly is strong evidence that it matters a lot. We know even more: N501Y probably increases transmission rates because it makes the virus better at attaching to our cells. Another mutation E484K is present in the South Africa and Brazil variants and is the most concerning of the mutations we understand. Among other properties, E484K appears to give the virus more protection against antibodies (our own or manufactured monoclonal antibodies). This makes our vaccines less effective against variants with this mutation and probably also makes reinfection with these variants more likely. This may be true of another mutation K417N as well, which is also present in both the South Africa and Brazil variants. It's possible that E484K spread quickly because it allowed variants with this mutation to reinfect people who had already had Covid from earlier variants (quite uncertain, under investigation). There are many more mutations that we don't understand quite as well as these three. They likely modify how the main mutations mentioned earlier work. For example, N501Y has arisen in some other variants that don't seem to spread faster - so it's possible that it needs to be combined with some other mutations to help the virus, or that the combination helps them a lot more than N501Y by itself. The opposite is also possible; some mutations may make N501Y less helpful to the virus. We still don't understand these combinations well, and they will matter in the near future.
7) I want to highlight the distinction between mutations and variants, and explain why we are worried about both. The same mutations can occur in multiple variants, either because it arose independently or because they inherited it from a common ancestor. The mutations I talked about above evolved independently in the UK, South Africa, and Brazil variants. Each of these variants also have other mutations that make them distinctive. The South Africa variant B1351 has the mutations N501Y, E484K, K417N and several more that we do not understand well. We keep track of which geographic regions some specific mutations are found because we know that they make the virus more dangerous to us. It is useful to know, for example, that E484K is found in most of the viruses in a country - people there may have less protection against the viruses they face. But because we don't understand many of the mutations well, we also need to track variants that are more dangerous; they may possess a particularly potent combination of mutations. For example, we know that the South Africa variant B1351 spreads faster and that we have less protection against it from vaccines, even if we don't fully understand all the mutations that cause that to happen (E484K is part of the story but probably not all of it). The next few months are going to involve plenty of discussion both of new mutations and new variants. There will be new variants that arise that have some of these same mutations as well as new ones. We will have to worry about these new variants, but we don't know how much to worry because we don't know about how the other mutations they will carry affect the virus. And there will also be new mutations that arise in some of these existing variants, which may lead to them being designated new variants - just like a single species of mammal can over time split into multiple species that we will give separate names.
8) Now that large numbers of people have been infected in many places, and we are ramping up vaccinations, many more people will have some immunity against the virus. This is a tremendous change in the evolutionary landscape. I expect to see the rise of a broader range of mutations & variants as a consequence. Why? Mutations that earlier may not have increased the spread of the virus much could now make them more successful (i.e. worse for us). Many mutations probably exist which earlier may not have conferred much benefit to the virus and may even have made the virus worse (which would be good for us, and would lead the mutation to die out). But if these mutations allow the virus to get around our antibodies (whether from a vaccine or previous infection) even partially, they now confer a huge advantage for the virus. Imagine a change to shape of the protein that the virus uses to attach to our cells. You could imagine a change that makes it slightly worse at attaching to our cells. Normally this would make it less infectious than background Covid strains and so viruses with this different protein would die out. But if the new protein shape is different enough that it is not 'recognised' by our antibodies, this mutation will now confer a huge advantage to the virus and will spread. It's possible that E484K spread because of this. It is not likely to be the only such mutation - we should expect to see many more in the coming months.
9) I want to reiterate that vaccination is going to massively improve our situation, and fast. Despite the spread of these variants, we should be able to get deaths down to negligible numbers in developed countries in months by doing this, and life will start to return to normal there. But developing countries will wait much longer for vaccination because of terrible planning: we have failed to build the manufacturing or distribution capacity needed to vaccinate a large chunk of the world's population quickly, and we should have. This is not even a moral argument. It is in the interest of rich countries to speed up vaccination in poor ones because large, partially-vaccinated populations will be a reservoir for evolution of variants that can return to rich countries and possibly evade their immunity then (again, this is fixable with updated vaccines, which can be designed quickly).
10) What can you do? Get vaccinated as soon as you can. Wear good masks (N95/KN95/FFP2) from now till at least a couple of weeks after your vaccination. Once you're vaccinated, you still might want to be a bit careful for a while. I will probably wear some mask in crowded or indoor spaces till cases are very low. But I think people who argue you should continue isolating as before even after vaccination are out of their minds. Go hug your friends and family once you've got some immunity. I sure as hell will.