In World of Warcraft slang (and perhaps in other MMOs too — I haven't played them), a gank is an unfair player kill. Originally, the term referred to being jumped by several players on the opposite side: the name was once a contraction of "gang kill."
What do I mean by unfair? Well, Warcraft has a system in place for rewarding player versus player combat, but it only grants rewards for being a good sport. Attacking and killing players far beneath your level is considered dishonorable by the game, and carries no rewards. Fighting other players within the honorable level range is totally straightforward, and doesn't carry the taint of ganking. But to beat up on the weak and puny is to be a ganker.
Ganking frequently occurs in low-level areas, where the lack of powerful players makes retribution difficult. The game has a system for alerting players when friendly towns are under attack, but the messages are also triggered by opposite-faction players accidentally coming within range of the town, with the result that many players turn the defense channels off.
Ganking is a relatively common occurrence on player versus player servers, where any player is open game for attack once they leave safe territory. Indeed, there are a fair number of players who take outright delight in starting and maintaining PVP characters specifically to harass low-level players on the opposite side.
Why would somebody take part in this? Well, it's because Blizzard did their job right. The Warcraft universe has two "good" sides (as opposed to the evil and neutral factions) — the Alliance and the Horde. A character belongs to one side, spends most of their time questing in service of their side, and the player develops a strong affinity with their comrades. Other than one-on-one duels, any fighting against other players is against the opposed faction, whether in Battlegrounds, the Arena, or simply in the world itself. The other side is wholly alien, too: when they speak, it's automatically translated into a nonsense-word representation of diagetic language: so to the average Human, an Orc and a Forsaken Undead would appear to be speaking very different languages, namely Orcish and Gutterspeak. /me commands, since a relatively early patch, have appeared to the opposite side as "Groovysnax makes some strange gestures," no matter what the originally command was.
The end product of all of this is that by the time you make it to a high level, you outright despise the other side. You've spent the past 40 levels running around areas where the Enemy lives in different towns, pursues different quests, and occasionally steals resources, chests, and rare monsters. This last is probably worse than anything else, because "ninjaing," as this behavior is called, can carry serious repercussions (like being banned from good guilds) if a player victimizes somebody on their side. An average player has a wide array of stereotypes and in-jokes, and likes nothing better to pound that dwarf's (or Tauren's, or whatever's) face into mush. The other side, despite the fact that they're basically the same, becomes the Other and is dehumanized.
So ganking is fun because you can justify it to yourself. When you nurture a festering dislike for the opposing alliance, it becomes much easier to divorce yourself from the fact that another player finds your ganking frustrating, to say the least. After all, if they speak Zandali and you speak Darnassian, the only way they're going to convince you to stop is to find powerful fighters of their own. And that's how arms races start.
Is there a solution to ganking? Not an easy one. Blizzard can't prevent it entirely without making the PVP system significantly more restricting. Players can move to a non-PVP server, at the expense of their guild and social circle. So until fun takes a backseat to courtesy, expect ganking to continue.
If all of this is confusingly technical to you, I've written up the slang of the Warcraft universe at World of Warcraft Slang.