A syndrome in which the subject experiences sexual desire for small children. Many cultures object strenuously to that sort of thing -- so strenuously, in fact, that it's often defined as a mental illness. The general view in those cultures is that the pedophile is incapable of forming romantic attachments in a normal way: He needs somebody he can completely dominate.
Shrewd observers of Western standards of beauty have noted that there's more than a little infantilization mixed in with our concept of a "sexually attractive" adult woman. Neotony (eye makeup, hm?), blah blah blah. In view of that, it seems both sad and ironic that some people also claim that pedophilia is linked to homosexuality. Projection, maybe? It's a fact that long eyelashes are a naturally occurring secondary sexual characteristic of adolescent boys. The girls gotta paint 'em on. Make of that what you will.
In many places, it's a very serious criminal offense to have sexual relations with children, though the legal definition of "child" varies a great deal.
Pedophilia is a more common form of the same term for the same phenomenon.