I would say the number one method is to give them a great obstacle that they deeply desire to overcome, then show us the story of them overcoming it.
There is also the "Pet the dog" or "Save the cat" scene where you see them doing something nice. Similarly, a character who loves someone is usually more sympathetic. If you have a guy who is a jerk to almost everyone, but he would do anything for his little sister, then he just got a lot more likable.
It also helps if the character has to deal with everyday crap that many people can relate to. The opening scene of Office Space comes to mind, where you immediately empathize with the character just by watching him wrestle with changing lanes in a traffic jam. Other common examples include getting dumped, having an annoying co-worker, being sexually frustrated, or an embarrassing moment like discovering an unzipped fly.
Make the character active. No one likes the damsel in distress. No one likes the character who whines and moans. The ones who stand idly by while bad things happen to other people are the worst. People love the character who gets up and does something about the situation.
Simple, but not easy: give the character a unique personality.