Forthwith is a old-fashioned, fancy way to say "immediately." An extremely obnoxious restaurant patron might demand, "Waiter, bring my soup forthwith!"
You're most likely to find the word forthwith in a book or movie set in the past, or in a legal document. Either way, it means "without delay." If you're summoned to the principal's office forthwith, you probably shouldn't stop to chat with your friends on the way. The meaning of this adverb evolved from an earlier sense of "together" or "at the same time," as in "going forth with another person."