There are general rules, like white with fish and chicken and red with beef and lamb but the rules are fraught with exceptions.
When you choose a white wine, the most obvious characteristic is its degree of sweetness, after that, the the varietal character of the particular grape and the amount of oak from the cask (if any) come next. Some whites are spicy, or flowery or both -- for instance a (dry) Riesling or Traminer, and pair well with spicy or Asian foods. Some hold up well against powerful fish dishes, like Chardonnay. Some whites are very versatile, and "suitable for all occasions," like pinot grigio, fume blanc, white burgundies, etc.
When you choose a red wine, the most obvious characteristics are the "amount of fruit," the "size" of the wine, and its age compared to its legs. Fruit means the degree to which you can sense the wine is made from grape juice. The counterpart to fruitiness are "spice," earthiness" and "herbal" notes. Size you may have heard of as "it's a big wine," this means the wine feels "round" and heavy in your mouth, and has a profound effect on your palate. A big wine may usually be complex, but it's presence is not subtle. "Legs" refers to the ability of a wine to age well. A wine with legs, served young can still be pleasant; but (a) needs a lot of aeration; and, (b) leaves the sense that as good as it is, it could still be better.
A really big wine is the star of the meal, and should be served with something simple. These include well-aged Carbernets, Pinot Noirs, Riojas, Rhones, etc. Juicy wines are good with simple foods too, but in a different way. Barbecue for instance. Some varietals that usually fit in this category are Zinfandel, Syrah, young Beaujolais, Chianti, etc.
The characteristics of the individual bottle are almost as pronounced as the characteristics of the broad variety. Your best bet is to use wine rating services or the advice of a good wine merchant. Try to bear in mind that most amateur wine "experts" haven't tried nearly as many wines as they say they have, and that most professionals and honest amateurs give the impression they know a lot more than they do. No one has the ability to sample more than a minisucle number of the nearly infinite possibilities.
If you don't know what you're up to, try and stay within the general guidelines. Get advice. Don't overspend, and don't cheap out -- price means a lot. Let surprise be part of the delight.