It is possible that alien language will be completely different from human languages. McKenna (1991) has suggested that aliens are already here and are already communicating with (some of) us: the aliens are psilocybin mushrooms and communication happens when we eat them.
Communication with aliens has been reported in accounts of UFO alien abductions. In the vast majority of cases, communication from alien to human is telepathic (e.g. Fuller, 1966, Jacobs, 1998). It is not clear whether the aliens understand spoken language; Jacobs argues that human-alien communication is also telepathic (http://www.ufoabduction.com/telepathy5.htm). Clearly, research in this area has only begun.
Science-fiction: Star Trek
Science-fiction writers often assume that at least some aliens will use ordinary human-type language, or languages that are easily translated into human language by translating devices.
The universal translator of Star-Trek has little trouble doing this, acquiring and translating at the same time. Its occasional problems and hesitations reveal that it operates on the principle of comprehensible input: the translator does not try to produce and then adjust its system when the communication fails (comprehensible output) nor does it get corrected. Rather, it listens and understands, and gradually acquires the system (see e.g. Star Trek Deep Space Nine, Episode 30: Sanctuary).
In general, Star Trek gets a mixed report card on language acquisition theory. In the first episode of the series Star Trek Enterprise, Ensign Sato was observed using a version of the audio-lingual method in teaching an alien language at Star Fleet academy (Star Trek Enterprise, Episode 1: Broken Bow). But in a subsequent episode, Sato presented a perfect portrayal of a Monitor over-user (Krashen, 1981), hesitant to speak without a firm conscious knowledge of the grammatical system of an alien language. Captain Archer persuaded her that the survival of the Enterprise was more important than the subtleties of the future tense.
Fuller, J. (1966) The Interrupted Journey. New York: Berkley Publishing Corporation.
Jacobs, D. (1998) The Threat. New York: Simon and Schuster.
McKenna, T. (1991) The Archaic Revival. New York: HarperCollins.