In Mass Effect 2, it shows each character's current loyalty. How do you increase their loyalty, and what does increasing it do?
Character Loyalty is something specific to Mass Effect 2. Getting a character's loyalty involves completing their character-specific Loyalty Mission, after which the character becomes "Loyal", and one of their powers becomes available to Sheppard as a bonus talent. (Also unlocks a bonus outfit, I think?)
Depending on your actions, you can also lose a character's Loyalty -- for example, Tali and Legion can get into a fight, and if both are loyal to you, the one who's side you don't take will lose their loyalty to you.
More information can be found here: http://masseffect.wikia.com/wiki/Loyalty
Raven Dreamer's answer is good, but as to the other part of your question (what does increasing [loyalty] do?), there's one thing s/he missed...loyal characters are more likely to survive throughout the game...which, in turn, improves the odds of your own survival.
Specifically, when you get to the final mission set you will encounter a series of choices that must be made about who will lead various side missions. Those choices, combined with which/whether characters are loyal, will determine whether everyone will make it back from the mission, or whether someone will die (permenantly). Characters that are not loyal are much more likely to perish. If too many of your crew die, (if you have less than two team members surviving to the end), you (Shepard) will not survive the final mission (though you may complete it successfully before your death, you will die in the aftermath.)
Having Shepard die during the final mission may seem like a non-event in a sense, since it happens after s/he has won the game...but it may well affect whether you can bring that save over to Mass Effect 3. (There is a loading screen tidbit that promises that you can import your ME2 character into ME3..."if you survive.")
One more point;
Zaeed, Thane, Tali and Samara
have missions that it is possible to complete, but fail to get their loyalty. So be sure to save before you try those.