He looks like a crudely drawn version of Patrick's face.
Noggin is drawn on Patrick's left hand by Plankton after Patrick wishes to have a friend to talk to while SpongeBob is at work using the magic bottle. Plankton, pretending to be a genie, tells SpongeBob and Patrick to both close their eyes while Plankton pulls out a black marker and draws a picture of Patrick's face. Patrick is excited and tells Noggin to say hello to SpongeBob because Patrick thinks Noggin is real and can talk. Patrick becomes more angry and because Noggin will not say anything, Patrick lets SpongeBob and Plankton go while Patrick is going to try and teach Noggin a lesson in his rock. Once Noggin and Patrick enter the rock, they are both heard fighting and loud noises are coming from the inside of the rock. Later in the episode, Patrick appears to be injured because of the fight they had in their rock previously.
Noggin is another term for the head.
Noggin is also the name of a spin-off Nickelodeonchannel that became Nick Jr. in 2009, and it's currently a subscribtion service since 2015.