While we can't speak much on the later events of "Urusei Yatsura," we can provide a summary of its setup. As mentioned previously, the show focuses on Ataru's adventures following his accidental proposal to Lum, an alien princess from outer space. Prior to that proposal, however, Ataru is just a pervy high schooler trying to reach third base with his girlfriend (via ANN).
When Lum's race of aliens visits and declares its intent to conquer Earth by winning a game, Ataru is chosen as the person to represent humanity in the aliens' traditional game of tag. His opponent, of course, is Lum. In order to motivate Ataru, his girlfriend promises to marry, and therefore sleep with him, after winning.
Ataru eventually defeats Lum in their game of tag, and excitedly expresses his enjoyment at the thought of getting married. Lum, however, takes this declaration as a proposal, and promptly agrees. Now stuck with an alien for a fiancée, Ataru finds himself involved in all manner of supernatural situations involving ghosts, aliens, and other such creatures.
