Brennan is a character played by Harry Goaz. Andy is a bit slow, even "dimwitted", and is deputy sheriff in Twin Peaks. He is very sensitive, and tends to cry at murder scenes. He is very loyal and trustworthy, and is compared to a dog by Albert Rosenfield. Andy has been seeing the secretary of sheriff's department, Lucy Moran. She, however has grown tired of him, and seeks adventure by seeing Dick Tremayne. Andy is initially jealous of Dick's relationship with his former girlfriend, however the two later bond over Dick's temporary foster son Nicky and end up becoming rather fond of each other. Not much is revealed of Andy during the show, except through physical comedy, such as his inadequacy at handling guns and sticky tape. However, Andy improves his gun skills at the range, later shooting Jacques Renault when he tries to go for Sheriff Truman.

Andy was the one who (in Season Two) realised that the drawing was a map, and he therefore played a big part in FBI agent Dale Cooper's finding his way to the Black Lodge.

In the pilot, he is revealed as a trumpeter, albeit not a very good one. And he has a talent at drawing which is demonstrated when he sketches Bob from Sarah Palmer's vision, and again during the trial of Leland Palmer when he sketches the back of Leland's head.