Dr. Drake Ramoray was a character played by Matt LeBlanc's character Joey on Friends. Dr. Ramoray was on Days of Our Lives, and he was in a coma for almost an entire season. Laying in a bed was an easy acting job for Joey, and it got him a spot on a major television program, despite his questionable acting abilities.
Eventually the writers decided to make Dr. Drake Ramoray a regular character. So he awoke from the "coma" and began interacting with all the other characters on the show. But it didn't last long, you see Joey was interviewed by Soap Opera Digest, and in the interview he claimed to write most of his own lines. Actually what happened was that he simply changed one of them once, and no one noticed. This angered the shows writers, and they promptly decided to kill him character off. Leaving poor Joey out of a job (again).
Joey briefly dated a woman (played by Brooke Shields), who could not separate fantasy from reality and thought that the Dr. Drake Ramoray character was a real person. Joey eventually pretended that he was actually the Doctor's evil twin ("Hans Ramoray"), and the girl promptly left him alone after that.
In general playing the Dr. Drake character gave Joey a big head and caused him to do all sorts of bonehead things, such as moving out of Chandler's apartment, and claiming to write most of his own lines on Days of Our Lives. He had to face quite a reality check after he was dropped from the show and had to go out and do auditions for commercials again. Where he soon found out that no one cared that he was Dr. Drake Ramoray.
Dr. Drake Ramoray references continued on the show long after Joey had lost the role. He absolutely loved it when people recognized him as "The Dr.". This role was how Joey learned his famous "smell the fart" technique for remembering lines. When you can't remember your lines, simply raise your head a bit like you may have smelled a fart, it ends up looking pensive and dramatic.