The single-player game puts you in control of Kane, a once-happy father who completely lost it after his son accidentally blew his brains out. So, like any grieving father would do, Kane toured the world as a mercenary until a super-huge job goes south... leaving everyone dead but him. Eventually arrested and put to trial, we meet Kane on a prison transport, sentenced to death. Until, that is, a group of people bust up the truck and set him free.
Lynch, on the other hand, is a schizophrenic crazy that's stuck in a constant blackout. His normal personality seems totally gone, leaving only a bloodthirsty thug who's also on the same prison transport, assigned to watch over Kane and make sure he plays nice with his liberators - who could be the very people who died during the botched job all those years ago.
So we've got one man who's spent his adult life killing and his ready to atone for his sins, and another who's totally off his rocker, frequently bashing people through the wall just for the sake of it. Lynch's unpredictable behavior often tramples on your ability to quietly perform your violent duties, so getting a second player to take the reins is probably a good idea. Plus, you get to watch each other's backs, trade off weapons and just be good all-around pals.
Later missions plop other mercs in your control, with basic squad tactics like "go here" and "cover me." If you played any of 2003's Freedom Fighters, you'll recognize the setup right away. Like Lynch, these guys have their own things to worry about and may even ditch the job once their goals have been met. Similarly, they may outright refuse to share weapons or revive you - Lynch in particular is not planning on letting anyone borrow his shotgun, so don't even ask.