When is SG used? When is OG used? When is FG used?
SG is used at different times of the brewing process. Extract brewers need to take less SG samples than all-grain brewers. Extract brewers typically only need to take one SG reading on brew day - the OG reading, which is taken after the boil process is complete. And even then, it's pretty optional. If you used all of the extract materials that you were supposed to and your volume is correct, then it's tough to have an OG to be something that it's not supposed to be.
All-grain brewers typically take an SG reading after the mashing process but before the boil, so they can verify that their SG is where it's supposed to be, and if it's not, then possibly take corrective action to adjust the wort. Then, after the boil, they take a second SG reading to again verify that things are where they should be, and again take corrective action if needed, and if everything is good to go, then that SG reading is the OG.
FG readings are taken after the fermentation process is complete, and for most beers, that process is usually completed within a week to ten days, usually shorter than that. And then they are taken three days later and compared to the previous FG reading to see if they are the same. If they are, then it's a pretty sure bet that the yeast are done reducing the specific gravity, and one can bottle up or keg the beer.
Keep in mind that even though the fermentation process is complete as far as the SG readings are concerned, this does not mean that the yeasts aren't doing other things to improve the taste / quality of the beer. Many homebrewers on here will tell you that it's a good idea to leave your beer sit and ferment for at least two or three weeks, to let the yeasts "clean up" certain compounds that may give your beer some unwanted off-flavors. Some beers really benefit from the extra time, some benefit from being bottled or kegged up fast and then drank very fresh. A big part of the art of homebrewing is figuring out when to do what to which beers.
