Vengeance Demons

And actually, we prefer 'justice demon'. Okay. FYI.
     - Halfrek, Older and Far Away.

The demons

Buffy has encountered two vengeance demons, although it's clearly stated that several others exist.
The first to appear on the show is Anyanka, the saint patron of scorned women. She is guided by Cordelia's desire for revenge over Xander and Willow. She creates an alternate universe, where Buffy never came to Sunnydale.
Anyanka became a vengeance demon after her boyfriend, Olaf, cheated on her. Having some knowledge of magic, she turned him into a troll.

Well he wasn't a troll then! You know, he was just a big dumb guy, and... well, you know, he cheated on me and I made him into a troll, which by the way is... how I got the job as a vengeance demon.
     - Anya, Triangle.

The second demon is called Halfrek, or Hallie, and has a thing about bad parents. She comes to Sunnydale for Anya and Xander's wedding, but decides to keep on working while she's in town, posing as the new guidance counselor at Dawn's school. Dawn wishes that the gang would be around her more and Halfrek curses all the people in the Summers home, gathered to celebrate Buffy's birthday, forcing them to stay there with Dawn. The curse affects their will, rather than their environment as Anyanka's spell did.
Little is known about Hallie's past, although she strongly reassembles Cecily, William's former love, but nothing has been clearly stated yet.

D'Hoffryn, a grey-ish demon with horns and pointy ears is a sort of master vengeance demon, Anya often refers to him as her boss. He's the one that gives them their powers. He also chooses who becomes one of his demons. He lives in the world of Arashmaharr, which is most likely also the home of all vengeance demons.
He clearly has a soft spot for Anyanka, attending her wedding and comforting her after Xander left her.
D'Hoffryn once offered Willow a job as a vengeance demon when she went magic crazed after Oz left her.

Their powers

Your powers were a gift of the lower beings.
     - D'Hoffryn, Doppelganglander.

Their main power is their ability to grant wishes. Vengeance demons take on a fake identity (like a high school student, a guidance counselor...) to get closer to their victim. They try to gain their trust and push them to express a wish.

Anya: Can't I just say... Men!
Cordelia: Second it!
Anya: Apart from being without class, the guy's obviously blind. Deserves whatever he gets.
Cordelia: I'm not even thinking about him. I am past it. I am living my life.
Anya: Still, I mean... Don't you kinda wish...

     - The Wish

Buffy: You didn't, by any chance, happen to... express like a, a wish, or-or something, to her?
Dawn: Um, ... maybe just a little.

     - Older and Far Away.

They also have great physical strength (Anyanka easily tosses Giles across the room in 'The Wish'). They are most obviously immortal, since Anyaka is over a thousand years old and Halfrek is approx 150. When hit by a sword, Halfrek isn't even slightly injured.

What, did you think I'd be stopped by a sword in the chest?
     - Halfrek, Older and Far Away.

Teleportation is another of their powers - Halfrek in 'Older and far Away', Anyaka in 'Two to go'.

Their demon status and powers seem to be related to their powercenter, that have the form of pendants. Anyanka's was green and Halfrek's is dark blue with little red flecks. Anyanka's was destroyed and yet she still could have her powers back and no longer seems to wear a pendant...

Their appearance

Vengeance demons have two appearances. One perfectly normal and human looking, another revealing their demon nature. it's only when they grant wishes or when they are among each other that their faces turn all wrinkly and raw-looking.

Summoning a vengeance demon

Vengeance demons generally sense the pain and the desire for revenge, as Anyanka did for Cordelia, but they can also be summoned. Giles does so in 'The Wish', mixing several plants and saying the following:

Oh Anyanka, I beseech thee. In the name of all women scorned. Come before me.'

And she appears in a mist of smoke. Anya summons Halfrek in 'Doublemeat Palace' to invite her to her wedding.

Halfrek: I have been called, and vengeance shall I wreak. Cower, masculine one... tremble as you face my wrath!

Anya uses quite another technique to summon her in 'Older and Far Away'

Hallie, get your ass down here now!

Reverting a vengeance curse

Giles reverts Anyanka's curse by destroying her powercenter in 'The Wish'. Hallie reverts her spell in 'Older and far Away', simply by saying the curse is lifted.
When one of Anya's spells causes the death of several young people, she begs D'Hoffryn to take it back, but that that means he must take the life of the demon to do so. Anya accepts, but D'Hoffryn takes Halfrek instead.

Becoming a vengeance demon does not heal the pain, it just makes the person relive it time after time forever. In Anya's case, it makes it practically impossible for her to actually have a real healthy relationship with a guy.