AVX2 is a 256 bit vector instruction set. You have 256 bit registers which can be interpreted several ways (8 floats, 4 doubles, 32 bytes, etc). AVX1 supports only floating point operations, AVX2 adds 256 bit integer operations. AVX-512 is a set of 512 bit vector instructions. There are only 2 flavors of AVX, plain old AVX and AVX2. AVX-512 comes in many different flavors. You may find Intel's Intrinsics Guide interesting.
The biggest difference is simply getting twice as many operations processed per instruction. Though, there are certain instructions in AVX-512 which may make some specific things more optimal (exponent approximations, for example).