Stark Tower was formally the main hub for most of the scientific and entrepreneurial endeavors of Stark Industries. Through an act of philanthropy Tony Stark donated the use the top floors to the Avengers, and had the space outfitted with all of the latest gadgets and security.
During the events of Civil War Tony Stark let Peter Parker and his family stay in the tower. After Peter turned against Tony and sided with Steve Rogers, he was kicked out. Later, Avengers Tower was used by Tony's Pro-Register Team, the Mighty Avengers.
After the tower's destruction during World War Hulk, Tony rebuilt it using S.H.I.E.L.D. funds. Tony was eventually removed from his position as Director of S.H.I.E.L.D. in the wake of the Secret Invasion, and replaced by Norman Osborn who took possession of all of Stark's belongings while reforming S.H.I.E.L.D. into H.A.M.M.E.R. (this included the tower, which he then made headquarters for his own Avengers team).
Soon after the fall of Norman Osborn and the Sentry during the siege on Asgard, Sentry's watch tower disappeared after his death. It had formally existed atop of Avengers Tower. Tony Stark was then given back the building, after which Thor placed a pillar on top to let the world know that both Earth (Midgard) and Asgard are allies.
The main tower is 93 stories high, and measures roughly 1130 feet in height (without including the now additional Asgardian pillars) The Avengers section of the building, also houses sufficient space for the many Avengers members, a communications room, hooked to various other important organizations, a general Avengers library (electronic), a forensics laboratory, a war room, a pathology research wing, and various hangars for various Quinjets and helicopters. The non related Avengers areas feature a children's hospital (Maria Stark Memorial Children's Hospital) and a substance abuse clinic.
In Other Media
In the 2012 film, at the end, Tony Stark's central hub that could provide renewable energy for all of New York City non-stop for a year was partially destroyed, only one letter remained: the letter "A." This was a nod to the fact that it will serve as the team's future base of operations.
Avengers: Age of Ultron
In the 2015 sequel, the rebuilt Stark Tower, now known as Avengers Tower, acts as the home base of the Avengers.
Avengers Tower appears as the team's base of operations in Avengers Assemble. The heroes first move into Avengers Tower at the end of the two-part series premier, with the building acting as their new HQ after Avengers Mansion is destroyed. It also appears in Ultimate Spider-Man, which is set in the same continuity.
In the MMOARPG game Marvel Heroes Avengers Tower (or Stark Tower) is the main headquarters of the players. There you can find vendors, healers and the storage system S.T.A.S.H. (Starktech Armory and Supply Hoard). Famous non-playable superheroes and other characters can be found within the building.