He is an alter-ego of Squidward with two different forms.
The first form, which requires a doctor to correct Squidward's facial injury, gives him a humanoid head. His body otherwise looks the same.
The second form, which requires Squidward to bash his head on the door after being in his first form, makes him appear muscular, as well as making his head and face even more humanoid. He also has acquired different clothing, which appears more like a leotard than a collared shirt. Overall, he appears to be what looks like an attractive man with six muscular tentacles.
Role in series
"The Two Faces of Squidward"
Squidward had become Handsome Squidward after hitting himself on the door. He became more handsome after a bit of doctor treatment. Handsome Squidward initially enjoys being handsome, but that is not long after he is constantly pestered by Bikini Bottomites about his "handsome" face.
Handsome Squidward soon hits himself on the door again, going into his second form, which is the more handsome form. He turns back to normal Squidward after getting hit on one of the Krusty Krab poles. Then along with going back to normal, Squidward is the same negative reputation of normal Squidward.
Squidward's second "handsome face" became a famous meme and running gag on the Internet.
It has also been used in the One Direction fandom, where a mathematical formula shows One Direction's Zayn Malik being added to Squidward's "even more handsome" stage, resulting in one of the guys from The Wanted.