Linus is an alien from the planet Linoleum who crashed into the legendary planet Earth. After returning home and expecting a hero's welcome, Linus soon found his fellow Linomen were skeptical of the existence of the so-called "planet Earth" and decides to return to it, this time with a camera. However, he is flat on the planets' currency, Linobucks, and must deal across the planet to get a new cosmic car and a camera. In his adventures, Linus deals himself out of Linoleum (including using a fake ID for Larry Flint to compete in a bumper car contest), quash a robot revolution in Detroitica and gets gas from an abandoned space station.
Each major location of Planet Linoleum has a teleporting device, which can be activated using a card. However, they often leave Linus with a side-effect, required to complete a puzzle. To travel between adventure sections, Linus goes through arcade sections, where he has to reach the other side of the level, avoiding free falls, enemies and collecting Cosmic candy at the same time (after collecting 10, an extra life is added). Linus dies at the smallest contact, so instead of speed running a level (which can be done, since the levels are small), it's more advisable to learn his foes' movement patterns and wait for a safe opening; this is particularly true in NES Linus Spacehead's Cosmic Crusade, where Linus can not change direction in mid-jump (he can in NES Cosmic Spacehead). Passwords are scattered at key locations.
All but NES Linus Spacehead's Cosmic Crusade include a two player mode named Pie Slap, reminiscent of Armor Ambush for the Atari 2600. While the Master System version is similar in graphics to the NES version, the gameplay is closer to the remaining versions. In the PC, Mega Drive/Genesis and Amiga versions, the art style is different from the NES version, much richer and closer to 60s cartoons like The Jetsons.
Source: Wikipedia, "Cosmic Spacehead", available under the CC-BY-SA License.