Broken, as far as Magic: the Gathering is concerned, is a term that is thrown around more and more frequently amongst competitive players these days. Simply put, a card that is broken is a card that breaks the format, or metagame. In essence, a broken card, or a broken deck, is one that is grossly over-powered in its current environment.
Just how broken a card is depends on a variety of factors, such as whether you are talking about limited (a format where you acquire the cards you play with and build a deck right before play begins, such as a sealed deck, or draft tournament), or constructed (a format where you build a deck from the cards you already own which are allowed, such as Type 1, Type 2, or Extended). For example certain cards would be very powerful in limited, where there are fewer forms of removal and the tempo of the games is slower, whereas in constructed the same cards could be considered nearly un-playable. A card like Living Hive is a good example, it is a very large creature that would win you many limited games when it came into play, but it's casting cost is too great to be considered for most constructed formats. By the same token, some card combinations that would be broken in Constructed would be totally unplayable in limited formats. For instance, Shared Fate and Leveler is a very potent combination of cards, but when they are not played in conjunction, they are very weak, so few people would attempt to play either of them in a limited format.
Some cards are so broken that they are banned or restricted from certain types of tournament play. The idea behind banning some cards in certain formats is generally that if left unchecked, decks based on these cards would become the only viable ones in a given format. A great example of a card banned for being too broken would be Tinker, which was banned from the extended format. Tinker was deemed too powerful, as it allowed players to play very high casing cost artifacts as early as turn 2 or 3, and it was a card that was showing up in virtually every successful extended deck until it's banning.
Having said all that, let me just summarize with a simple breakdown of broken: Broken means good. If someone refers to a card as broken, expect to see it the next time you go to a tournament, and if you want to win, use cards designed to deal with it.