source mameaddicts by HierophantIt's a good idea to check the Save State folder in the left field of the MAME GUI first for games with full save state support. Other games that are not in that folder may still be able to do save states, although in this case the stability of the function is not guaranteed.
The defaults are Left Shift+F7 to save and F7 to load. When calling up these functions you then need to pick numbers or letters on the keyboard as positions to save to or load from. For example, you play for a while and decide to save>go Left Shift+F7 then press 1 to save to position 1>you want to continue later>go F7 then press 1 to load from position 1 and get back to where you were. You can keep adding new save/load positions on other keys as you progress through a game to build up a collection of handy save points, or if you're not fussed you can just keep overwriting the same position.
Saved states will automatically go in the sta folder of the MAME root directory if you ever want to clear anything out.