How do you increase your team members' loyalty in Mass Effect 2?

In Mass Effect 2, it shows each character's current loyalty. How do you increase their loyalty, and what does increasing it do?

From gaming sirconnorstack

• 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

  Andy : The character's loyalty mission becomes available to you at a certain point in the game automatically. It's not like you have to search for it.

  Zemm : I'd also note that you can complete their mission, and not gain their loyalty, based on the WAY you complete their mission. Also to note, when you gain their loyalty you unlock a power for the character, which is also available to Sheppard in his extra power slot (the only one you get to choose, and you only get to have one at a time).

  Zemm : @Andy: well it's not quite automatic, as you can completely avoid talking to the characters between main story missions and never pick up their loyalty mission. But they're certainly not hard to find either, your secretary will tell you to go talk to so-and-so when so-and-so's loyalty mission is available whenever you approach the galaxy map.

  Andy : @Zemm good point. I just meant that it's not like you have to know to travel to a certain planet to find the mission - the game makes it pretty easy to make sure that you know you have loyalty missions to do.

  BlueRaja - Danny Pflughoeft : Also note (SPOILER) that if you pick a teammate who is not loyal for one of the tasks in the final mission, they will die, even if you complete the task perfectly.

  Raven Dreamer : It's a little more complicated than that, BlueRaja, but the intent is correct.

  Donal Fellows : When you've got a choice between retaining the loyalty of one of two team members, you can keep both loyal if you can pass a paragon or renegade check. This in turn requires doing certain loyalty missions quite late in the game (naturally that's not so much of a problem for Legion's case).
  From Raven Dreamer

• ---SPOILERS COMING---

  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.")

  rmontagud : Beware also with the fights between team mebers, namely Miranda/Jack and Tali/Legion if you don't have the proper Paragon/Rebel points you won't be able to convince them to stop the fight and you will have to choose between one of them getting downgrading the loyalty of the "loser" in that fight

  Beska : @rmontagud: Quite true. Apparently, you can eventually repair the damage through talking to them again later, but it takes a *much* higher paragon/renagade score to do this than it would have to just get it right in the first place.
  From Beska