What is 64-bit Software?

I know that computer operating systems can be 32-bit or 64-bit. I know that 32-bit software can run on both 32-bit and 64-bit systems but 64-bit software can only run on 32-bit systems. My questions are:

1) What exactly is 64-bit software?

2) Whats the difference between 32-bit and 64-bit software?

3) What are the benefits of writing 64-bit software on 64-bit systems instead of just writing 32-bit software for both systems?

Thanks