Beat Up is an attack like a multi-hit attack (Rock Blast, Fury Swipes) that hits how many times you have a healthy pokemon. The power for each hit is 10, iirc. It is neutral. Beat Up uses the pokemon's base attack and the opponent's base defense. Fainted or statused pokemon do not have their base attack hit included in the overall hit.
One example is houndoom. Blissey is a common switch in being a special sponge. If houndoom uses beat up, then the base attack of itsself and its partners will damage blissey according to her base defense. Since her base defense is quite low, she will get 2hko'd most of the time, resulting in a sweepage. Other than that, it doesn't have much use except for hopefully breaking a sub.
Here is another explanation that might be clearer posted by Obi at smogon:
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It uses the regular damage formula for each hit. It does a base 10 power typless (so no STAB, no effectiveness changes) attack using your base attack instead of your normal attack, and it uses your base defense instead of your normal defense (so if you hit Celebi with Beat Up, it uses 100 for the defense value every time, rather than 236, 299, 328, or whatever). This is why it's very effective on Blissey, as Blissey has a pathetic base defense, but uses EVs to nearly triple its defense (compare 10 base defense with the usual 130 defense stat to see why). It does a base 10 attack for all your Pokemon that have no status conditions and are alive, using their base attacks each time.