The elevator refers to the elevator in the apartment building where Sheldon, Leonard and Penny live. It has been out of order since 2003, four years before the series began, as told in "The Staircase Implementation" (S3E22) although Leonard once said two years previously in "The Nerdvana Annihilation" (S1E14). It has been broken ever since Leonard tested out an experimental rocket fuel formula which began to explode, with Sheldon saving Leonard's life after he tried to take the fuel outside by going down the elevator, but was stopped by Sheldon closing the elevator after taking Leonard out, in "The Staircase Implementation".
The Apartment's elevator was working in good order in 2003. Residents and visitors use it to go up the floor, rather than the stairs. When Leonard moves in the apartment building, Sebastian exits the elevator, carrying his stuff. Before leaving, Sebastian warns Leonard about Sheldon leaving Leonard confused. Leonard uses the elevator to get to the third floor where Sheldon's apartment is located.
When Leonard does an experiment of an rocket fuel formula, and during an brief argument with Sheldon, the fuel begins to go awry and building up pressure which Leonard tries to get it out of the apartment before it explodes. Sheldon drags Leonard out of the elevator, pushes the button and jumps out, leaving the fuel inside. The rocket fuel then explodes, destroying the elevator, leaving just the doors and the elevator shaft intact and unharmed. After the incident with the elevator and saving Leonard's life, Sheldon didn't report him to the landlord, the police, nor homeland security.
Following the incident with the rocket fuel, the elevator stays defunct, meaning that the residents and visitors will now have to use the stairs to go up and down floors.
Instances of opening the elevator
In "The Nerdvana Annihilation", Leonard was dreaming about helping Penny get down the stairs when their time machine was blocking them. He ripped the warning signs off the elevator door, opened the door, grabbed the elevator cable, took Penny in his arms and descended on the cable while Penny kissed him as her hero.
In "The Jiminy Conjecture", Raj, Howard and Sheldon are searching for a cricket they had heard. Sheldon searches the bottom of the elevator shaft. After his friends run away after hearing the cricket, Sheldon is left in the elevator shaft in the dark when his flashlight gives out.
In "The Lunar Excitation", a drunk Leonard pushed the elevator door open a little and drop a wine bottle down, and from the time before the bottle hit the ground, he evaluated the height of the fourth floor is "thirty feet".
In "The Scavenger Vortex", Leonard pushed open the elevator door and searches the elevator shaft for a clue in the scavenger hunt that Raj planned.
In "The Holiday Summation", It is mentioned that Leonard and Penny had placed a chopped down Christmas tree inside the elevator shaft after discovering a squirrel is in it.