This is not my area of expertise, but I think I can help answer the question. Everything mutates a lot, so there is a lot of genetic variability. These are usually pretty minor, but really help with tracing, as you can correlate "family trees" of mutations. The big one early in the pandemic was you could see the difference in the genome between the China/West Coast USA and Europe/East Coast USA infections.
I'm not sure of the strict scholastic definition of a strain, but I think they're derived from branches of that family tree. Most of those branches behave pretty similarly as they have mutations that aren't really impactful, but some of those behave differently, and those are the "strains/variants" we're talking about.
On an interesting note on how genetics are used to target therapies, what researchers can do is look at all the different mutations and they can find parts of the genome that change a lot, and parts that change very little. The parts that change very little are often critical for the function of the virus. Mutations to those still happen, but because of the critical nature of the part that mutated, the virus isn't able to reproduce as effectively and so doesn't show up as often. Researchers can use those as targets for treatments, as disrupting those can impact the effectiveness of the virus.
If I got any of that wrong (or I'm using the wrong vocabulary), I'd love to hear from someone with an actual background in this stuff.