Wilmer is Gutman's punk kid enforcer.
He is young and completely out-gunned and out-classed by Spade. We don't know much about Wilmer, except that he's steadfastly loyal to Gutman up until the moment Gutman agrees to turn him over to the police. Even though Wilmer likely sees Gutman as a father figure, he cannot forgive Gutman for betraying him and kills him in the end. Wilmer is also violent, hot-tempered, and prone to being trigger-happy (he's killed three people in the novel that we know of: Thursby, Captain Jacobi, and Gutman).
The narrator also hints at the fact that Wilmer is Cairo's gay lover. Wilmer's delicate eyelashes and pale skin disturb Spade so much that he never misses an opportunity to insult or even assault the young man. As a homosexual couple, Wilmer and Cairo stand in stark contrast to the hyper-masculine, hardboiled tough guys like Sam Spade. For more on Hammett's portrayal of homosexuality, take a peek at Joel Cairo's "Character Analysis."