The Lair is an old, abandoned subway station under the streets of Manhattan. This was the hideout and headquarters of the Tomorrow People.
They have a computer named TIM which has an artificial intelligence. This was stolen from Ultra by John Young.
Another feature of the Lair is a boxing ring used to train in fighting techniques as well as a means of settling disputes via "jousting".
It can be assumed that the Lair is the abandoned City Hall Subway Station beneath Manhattan