Each puzzle is a single file on your disk. Each game is also a single file.
You can locate the folder of the file corresponding to a thumbnail in the gallery: right click on a thumbnail and it will appear a pop-up menu with that option.
When you start playing a puzzle, BrainsBreaker automatically creates a file to store the state of that game.
You can stop the program in the middle of a puzzle and it will resume when you restart it.
If you leave the program while it shows a puzzle, the next time you restart it will show that same puzzle as you left it. No need to worry about saving and filenames if you don't want to. (If you leave the program while it shows the gallery, the program will show the gallery at restart).