I've been playing Skyrim on my brother's Steam account, and decided to make my own account. I made it possible to play some of his games on my new account, but if I buy Skyrim so that I have it on my account, will the save files be the same?
You can indeed, they are just files. You WILL have to pick and choose your files to get the ones that are yours but by default Skyrim stores the save game files at this path:
Now clearly you are going to have to replace the James part with the name you use on the computer but its just that simple. Basically the save files are in the Documents folder\my games\skyrim\saves.
Yes. The save files are independent of your steam account. As James has mentioned, your saves are stored in your "Documents\my games\skyrim\saves" folder.
Let's say you and your brother each have your own computers, but you were playing games on the same account (that of your brother). Then your brother will have his own saves on his computer that are not visible to you and your saves will not be visible on his computer when he's playing Skyrim.