There's a thread over at EN World where a poster asks what "swingy" means in terms of D&D. Here's the definition for it that I use:
A swingy system is one where a single bad or good die roll dictates the outcome of an encounter for a player or the entire group. If you roll a 1 on your save against a ghoul's paralysis, you're out of the fight unless the cleric has the right spell to get you back on your feet. As with almost everything in RPG design, a swingy mechanic has its strengths and weaknesses.
The biggest problem with a swingy game is that it produces outcomes that crowd out the rest of the system. 3e had a huge emphasis on pre-fight buffs and spells taken specifically to counter rare but catastrophic outcomes. It really favored players, because it was far more likely that they could plan and optimize to take advantage of the system's swing rather than fall victim to it. The reverse was true for DMs, who have multiple critters to manage and villains to create from scratch.
However, swing has a huge benefit. It makes every die roll tense and dramatic. It drives players to the edge of their seats and keeps them there. If you look at 4e, with its efforts to reduce swing, you see a common criticism of overly long fights and battles that are decided long before they finish. Some of that might come from encounter design, but at its root the critics are pointing out the lack of swing in the game.
Personally, I like tense die rolls, but I don't want all my rolls to have that tension. For instance, gnolls in 4e get a big damage bonus if they gang up on a PC. In one fight in my Temple of Elemental Evil game, the gnolls spread out to engage the characters. Once the fighter was bloodied, though, the gnolls used their next turn to move over to him (taking opportunity attacks along the way) and wallop him from barely bloodied to below 0 hit points. That had exactly the tension I like. When the first gnoll hit for a lot more damage than normal, the players all leaned in to watch the next two attack rolls. In 4e, that swing is moved from a single die roll (Save or die!) to a series of die rolls (If two of these three gnolls hit the fighter, or if you blow your next two saving throws, it's lights out).
Oddly enough, while balance and swing may seem like polar opposites they go hand in hand. A perfectly balanced system would be dull and a wildly unbalanced one leaves players and DMs essentially inventing their own game. By the same token, a wildly swingy game might as well use a coin flip to resolve entire battles, while one without any swing is as fun as watching paint dry. Both are boring.
Balance establishes a level playing field between characters and gives a DM a clear sense of a critter's strength compared to the PC. Swing steps in and throws all that stuff out of whack on a session by session basis. It determines that in this fight, the rogue is the hero who leaps over the crowd of orcs and skewers the evil priest with his rapier, while the fighter stumbles around with a dark mantle stuck to his head. It lets the 6 Intelligence cleric spout out the answer to the Lorekeeper's question on arcane theory while the wizard sputters and flails. It makes for memorable moments, because it produce rare, interesting, and big moments.
Balance sets the stage for swing. In a well-designed game, the players cannot control or eliminate swing. It's the wild card that reminds them that the best laid plans sometimes fall apart, while even the most hopeless struggle can turn on a die roll or two. It's the unpredictable element that makes games exciting.