A delta wave was a brain pattern usually associated with deep sleep or a coma.
When Doctor Julian Bashir was attacked by a Lethean named Altovar in 2371, he fell into a coma. He believed himself to be conscious and on Deep Space 9 after the attack; however, he learned that what seemed to be the space station was actually a representation of his mind, since a tricorder he used was only able to pick up delta waves from himself and nothing else. (DS9: "Distant Voices")
According to an ultimately unused line of dialogue from the final draft script of TOS: "Dagger of the Mind", a patient's delta waves and brain pattern were typically monitored by a particular index on a body function panel in the sickbay of the USS Enterprise.