Companions are found in various locations in the game. Some may have a slight requirement before you can recruit them...a simple quest or skill check passes, or certain faction rep checks or just in difficult locations to reach. Some can be recruited as soon as you meet them. They're just npc's you encounter in the game, and if you go through their dialogues you'll either be able to recruit them right during, or you'll get a quest and when it's done, the dialogue to recruit them will open up.
You can have 1 human and 1 non-human companion at a time, so if you want to try a different one, you have to 'fire' one first. The first companion a new player is likely to encounter is Boone, and he's in Novac. The easiest thing to do is direct you to the fallout wiki page: http://fallout.wikia.com/wiki/Fallout:_New_Vegas_companions