It means that a soldier got separated from his comrades during a battle; he may have been killed, or taken prisoner, or be lying injured somewhere, but nobody knows for sure.
Sometimes people use it in more everyday contexts, although it's slightly humorous. It's quite common on internet message boards, for example, when someone who has been a member for a long time suddenly stops posting. One of the other members might make a comment about that person being "MIA": perhaps he's got too much work to do, perhaps his computer's broken, perhaps he's ill -- nobody knows.
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