A motherboard is usually attached to the case using a combination of standard and case mounting screws. Mounting screws are similar to those male-female adapters, and have a threaded screw at one end and a recessed input area on the other. The motherboard sits on top of the mounting screws, while standard screws are inserted above the board and attach it firmly to the case. This format is used to raise the motherboard off of the metal case, which if not done, can cause the motherboard to short, malfunction and can even cause scoring on the case itself. Some cases also feature raised mounts that serve the same purpose as mounting screws, but are usually larger and a bit more stable.
Close-up of mounting screws on back of case
Since the new motherboard may be a slightly different size than the old one, you need to compare the new board to the case or to the previous motherboard, and see if that the mounting screws line up. If they are the same, no changes will need to be made. If it turns out they are different designs, then line up the board to the case and simply change the location of the mounting screws. Some cases have schematics printed on the inside of the case, and even if your case is bare there are really only a few possible ATX combinations to deal with. Those adding a motherboard to a fresh case will be starting with a blank slate, so mounting screws (included with new cases) will need to be attached using the motherboard layout as a reference.
This is also a good time to look at the ATX backplate on the case itself and ensure that it matches the new motherboard. If you are upgrading from a standard board to one with on-board sound or other options, then chances are you will have to pop a few new holes in the ATX plate. Performing a quick check beforehand saves time and hassle over finding problems only after the motherboard has been partially inserted.