A "broken" card is one that is commonly considered to be exceedingly over-powered, or has an ability that is inherently degenerate, and has a large game impact. The superlative form is called Bah-roken, which was coined by William Jockusch.
Most cards that are widely considered broken are components of oppressive (a.k.a degenerate) combos which within their format can win too quickly or too reliably. Format certainly does affect how broken a card is perceived as being. In legacy or vintage, recent cards are very seldom broken because they are up against spells like Force Of Will which are significantly more powerful than could be printed now without themselves being broken.
Broken or not?
Controversy arises when broken is used too often, however. Because of its subjective definition, there are great debates over its application to some cards. For example, many would call Skullclamp "broken" while others would say it is simply overpowered. Cards that are genuinely broken tend to radically warp the metagame around themselves, where the vast majority of players either play using those cards, or decks that are specifically designed to beat only one single deck. If cards do eventually have this effect, then it is likely that the cards will be banned or restricted in order to revitalize the metagame and return to a less predictable and enjoyable game.
Cards on which there is a general consensus that they were broken (and which were subsequently banned at least in certain formats):
- The Power Nine
- Tolarian Academy
- Sword Of The Meek as part of a combo with Thopter Foundry which could be used with a wide variety of non-broken pieces to gain infinite life, deal infinite damage or similar.
- Stoneforge Mystic which allowed faster use of extremely powerful equipments with very high reliability to the point where a deck only needed a single copy of three different artifacts.