The Riddler's lair is the personal headquarters in which The Riddler resided and worked.
Arkham Origins Incident
While Batman did not meet Enigma in person, he did find the room where Enigma held his servers, and was ready to release the extortion data the following day. Enigma's Headquarters could be find in the Burnley District, behind a billboard. It had several rooms out of which two were accessible, due to the elevator only stopping at the lowest level. The first room was flooding at places and included a large computer that was protected with a electrical field, which prevented Batman from destroying it before all the network relays were taken down.
One Extortion Pack could be find on a wall over the water. The second room included a jammer, which prevented Batman from hacking the server, and could only be accessed after he collected all of the extortion data. A hidden room could be found from the second room, by breaking through a certain wall. Behind that wall, the very first Riddler Trophy could be found and collected.
Arkham Asylum Incident
Arkham City Incident
After he learned that Batman was in Arkham City, Riddler kidnapped the medical team from the Church and Medical Center. He locked five of them in his Riddler Rooms that were created to challenge Batman and let him and the hostage die. Riddler took the rest to his own room, where he could monitor the other rooms and make the hostages walk around his base. If they stopped for any reason, Riddler would blow them up.
Arkham Knight Incident
After being released due to Protocol 10 violating the rights of the inmates, Riddler started hiding on an abandoned orphanage, that was located on Miagani Island. Riddler purchased several unused sewer and underground areas from the city, where he would build his Racetracks and Puzzles. Riddler himself resided in a hidden room behind a bookshelf in a hidden wing inside the orphanage, from where he presumably sent his broadcasts. Riddler also had a workshop, located below Winslow's Toy Shop, where he built his army of Riddler Robots, and according to himself, a "Riddler Mobile" which was apparently much better and faster than the Batmobile. This machine presumably didn't exist.
Most of the orphanage was turned into his personal playhouse, with most of the rooms turned into physical challenges, that were meant for Batman and Catwoman. The main hall was fitted with special flooring, which could be electrified and lowered when needed. After completing his final challenge, Riddler himself showed up, fought Batman and Catwoman, and hid under this flooring, up until Batman solved all of his riddles. He was then defeated by Batman and Catwoman and was delivered to the GCPD Lockup.
It is unclear where Riddler resided during the orphanage events; whether he hid in the inaccesible wing west of the player when entering the orphanage. He may have hid under the floor as seen in the fight between Batman, Catwoman and Riddler. It is possible he does not reside in the orphanage at all. He may have also settled in the hidden room found southeast of the building.
Catwoman's Revenge Incident
Riddler had an another hideout at Winslow's Toy Shop that he had taken over. After the events of Batman: Arkham Knight, Catwoman planned to exact revenge on Riddler by sneaking in and robbing the place, while he was still in prison. She succeeded, initiated the factory's self-destruct mechanics, and left Riddler without his money or his army.