Azazel served Sebastian Shaw in the Hellfire Club. He killed The Man in Black and many other CIA agents with Riptide and Shaw during their assault on the Division X facility. They attempted to recruit the newly-formed X-Men into the Hellfire Club, and was able to bring one of them to their side: Angel Salvadore. His hijacking of a Soviet freighter during the Cuban Missile Crisis nearly led to an all-out nuclear war. After Shaw died, Azazel and the rest of the Hellfire Club joined the Brotherhood of Mutants.
In July 1963, Azazel and Angel Salvadore were slain by Project WideAwake operatives in a firefight. They were ambushed by them while they were asleep. Afterwards, their bodies were autopsied and examined by Trask Industries.
Powers
Teleportation - He is able to psionically transport himself, his clothing, and additional mass across great distances, giving him the appearance of being vaporized into a very brief puff of red smoke or gas, which appears where ever he teleports to or from.
Prehensile Tail - Like Nightcrawler's tail, it allows him to grab onto objects with relative ease. His tail is strong enough to not only support his body weight but also lift an adult man completely off the ground. It is deft enough to fight with like a sword or blunt object and its talon or stinger like tip can be used as a stabbing weapon.
Superhuman Athleticism - Azazel's agility, balance, and bodily coordination are enhanced to levels beyond the human body's natural limits. Azazel's teleports can weaken those who are not teleporters as his body is naturally resilient to his own teleports. Continuously teleporting with passengers will slowly weaken the passengers. He was also capable of taking a few blows from the super humanly strong Beast.
Low-Level Healing - Azazel can heal minor injuries with ease, however he still retains a scar across his face.
Retarded Aging - Azazel possesses slow aging; it is unknown how old he truly is. His comic counterpart is over 1,000 years old. He may have met Sebastian Shaw before the events of WWII and is very likely his first and oldest ally as the both have a means of aging slowly.
Enhance Reflexes - Azazel can react to gunfire almost immediately, he can decisively teleport and/or dodge around any battle field. His reflexes match Beasts and he easily avoid Havok's beams twice, despite the speed of the beam being light speed.
Extrasensory Ability - Azazel does not seem to have a limit to how far he can teleport, however it is likely when Azazel teleports into locations he has no knowledge/memory of. He possesses a latent psychic ability to carefully teleport without re-appearing within a solid object. Azazel can predict other mutant teleports if they are near him.
Abilities
Master Swordsman - during his assault on the C.I.A, Azazel is seen wielding two swords which he uses to kill many of the agents
Master Martial Artist - Azazel is considerably the best martial artist within the Hellfire Club. He also managed to easily hold his own against Beast and Havok simultaneously and was only defeated by Mystique's distraction. It is possible he trained Mystique between the events of X-Men: First Class and X-Men: Days of Future Past as she did not possess any acrobatic or martial arts ability until after joining the Brotherhood.
In the comics he is the father of Nightcrawler. He had Nightcrawler with Mystique. It is unknown if this will be explored in the films.
In the comics, Azazel is a mutant of demonic descent, and was born in Ancient Mesopotamia. In the movie he appears to be a Russian human mutant with demonic traits.
When teleporting he leaves a fire-like trail behind, while his alleged son Nightcrawler leaves a smoke-like trail.
Though not considered to be canon, the X-Men Origins: Wolverine videogame reveals that it is in fact John Wraith, rather than Azazel, who is Mystique's mate and Nightcrawler's father.
Actor Jason Flemying is actually British, despite playing a Russian character.
Next to Apocalypse, Azazel is one of the oldest mutants to live in the X-Men universe.
Azazel is not a member of the Hellfire club in the comics.
Azazel on the Marvel Wiki
Azazel on Wikipedia