A type of wood, used to make barrels that wine may be fermented and/or aged in.
If tannin was the first piece of wine lingo you learned, then the word oaky likely came shambling in right behind.
(There's a good chance the word buttery was uttered in an adjacent breath, and, if so, the topic was probably the grape called Chardonnay. Let's just go ahead and unhinge that association. If you need help with this task, go to the French section of your local wine shop and buy a bottle of Chablis. Drink it, and keep reminding yourself that this slender dude is Chardonnay, too. It will be unbuttered and oakless. May your horizon be permanently stretched.)
Most commonly, when we use the word oak, we mean to describe the flavor that new (or nearly new) barrels bestow upon wine that has spent time inside them. This flavor can vary, but usually anything that smells like coconut, vanilla, cedar, cloves, nutmeg, cinnamon, or, well, the split wood of oak trees, is a dead giveaway for aging in new (or newish) oak barrels.
Uncertain? Go pour yourself a glass of bourbon, and smell that. There's what oh-so-much oak smells like. Whole splintery planks of it. For me, it took being in a roomful of new barrels, all filled with a modern-styled Barbera d'Asti. The air rang with the sweet scent of resinous new wood. I never again tasted that wine in quite the same way as I had before. Or any other wine, for that matter.
After a barrel has had wine in it for a couple of vintages, it stops giving off much oak flavor. We call these barrels neutral, and the distinction here is crucial. If a wine only sees neutral oak, it won't taste oaky. Many wines see (that is: are aged in) a mixture of new and old barrels, and the higher their proportion of new barrels (and the longer the wine spends in them), the more oak flavor the finished wine will end up having. Oak from different countries, forests, and even specific coopers will impact flavor, also. American oak, for example, tends to give lots of coconut and vanilla flavor, while the French stuff tends to pass on more prickly tannins.
Oak flavor isn't the main reason why wine goes into oak barrels, however. Oak is a porous wood, and wine slowly evaporates out, creating a vacuum inside the barrel. This vacuum sucks outside air into the barrel through those same pores, oxygenating the wine at a microscopic rate. This subtle influx of oxygen helps nurture increasingly complex flavor and aroma compounds, and it keeps the wine from creating odd off-flavors, like those of rotten eggs, a problem called reduction (more on that some other time, I promise). The size of the barrel, along with the tightness of its grain, will decide how fast or how slow the process goes. And since the grain of every tree is different, each barrel ends up slightly different, too.
New oak has become a bit of a divisive issue, with some sommeliers and winemakers having strong objections against its obvious use in a wine. New oak can lend complexity and texture to a wine, particularly as it mellows with age, but it does seem clear that there are diminishing returns beyond a certain point. Some wines—like traditional Rioja from Spain—have oakiness as a classic, traditional part of their flavor profile, but for others—Italian reds, for instance—oak flavor is a modern garnish. Finding that line of appropriateness is tricky, and ultimately falls to good ol' unaccountable taste.