Oak can enhance the color of the wine, soften and round out the tannins, and impart its own unique characteristics. Almost all red wines and many white wines spend time in oak barrels before being bottled, and that's just because winemakers have found they taste better that way.
When a wine sits in oak to age, the oak slowly imparts its flavors and colors into the wine. If this is a white wine, the longer the wine sits in oak, the darker a yellow it will become, almost mimicking the hue of straw. If the wine is red, color is not affected as much, but often the longer the wine sits in oak, the darker red it can become.
In terms of flavors, living inside oak is a compound known as vanillin, which as the name suggests, tastes like vanilla. When a wine sits in oak for a long time, that compound leaves the wood and transfers into the wine, which is why many white wines, especially Chardonnay, can have such prominent vanilla flavors. Other flavors that can be enhanced by oak are mocha, caramel, toffee or honey.