Doom Patrol was about a super-team composed of outcasts with incredible abilities. Pilot Larry Trainor, exposed to strange rays while in flight, could release (for a very short period of time) a purely energy being known as Radioactive Man. Rita Farr was the Elasti-Girl, who could change the size of her body, or any single part of her body. Cliff Steele, his body destroyed in an accident, had his brain implanted into a super-strong robotic body, to become Robotman. (There was another Robotman who featured in backup strips in other DC comics.) Leading them was the Chief, Niles Caulder, who supplied them with their headquarters, funding, and various gadgetry. He himself was confined to a wheelchair, but had specialized chairs made that contained various gadgets and weapons, such as a large iron claw on the end of an extending arm, etc.
Facing our team was a weird cadre of villains, led by a recurring badguy group called the Brotherhood of Evil, who were led by a brain in a jar and included a talking gorilla and a French villainess who could change her features. A genius called Mento created a helmet to give himself mental powers, but he couldn't bring himself to destroy the team, as he was in love with Elasti-Girl. Beast Boy was a green-skinned teenager who could assume the shape and abilities of any animal; all he wanted to do was join the team because he was an outcast, too. (He would later re-emerge as The Changeling, a member of the popular New Teen Titans from the 1980's.)
Doom Patrol was one of the most offbeat superhero comics of the 1960's, especially considering it came from DC, who at that time were still mostly jacket-and-tie types compared to other companies. The characters bickered among themselves (as the fledgling Marvel heroes did all the time), stories were full of little jokes and wisecracks, and the team's adventures couldn't possibly be taken too seriously. The attitude came off as being one of unadulterated super-heroic fun. Readers may have been put off by the strangeness of some of the characters and goings-on, but the writing (as well as the art style, which stuck to the milder DC standard of the era) soon put them at their ease. Until a later Doom Patrol series came about in 1987, the group was largely forgotten, or at least went unmentioned, for several years.
The Doom Patrol first appeared in the sci-fi anthology series My Greatest Adventure in issue #80, and were so popular that the title was changed with #86. The series was a solid seller for some time, but ended with #121, wherein the team were killed while trying to save a small town. Although the editor and artist of the title themselves appeared in the issue to ask readers to voice their support for a possible return, the title would remain dormant except for a handful of reprint issues five years later.
You can see a gallery of the Doom Patrol covers by going here.