A frying pan, also called a skillet, is a pan, usually round, with low, gently sloping sides and a long handle. It's used, as its name suggests, for frying foods, usually over high heat, so it needs to be thick enough not to warp. A good heavy frying pan will hold its shape, and the better it is, the more likely it is to conduct heat evenly, without hot spots that will burn food.
Most frying pans are round, usually 8, 10, and 12 inches in diameter, though electric frying pans may be square or oblong. I find it useful to have a lid that fits my frying pans, and an oven-proof handle is nice too, allowing you to move food from stovetop to oven easily. This is handy both for baking and broiling foods; you can try wrapping a plastic or wooden handle in tin foil if it isn't going to be in the oven long. Some people find stay-cool handles a boon, but I tend to stick with metal handles and use a potholder to pick up the pan.
A frying pan is not the same as a sauté pan, which, as the good LordOmar points out elsewhere, has straight, not flared, sides. If you can afford it, have both, because each is specially designed to fulfill a different task.
Frying pans are made for pan frying (not deep frying or stir frying), and the sloped sides help ensure that steam does not collect in the pan. They are perfect for making omelettes (although true omelette pans have slightly rounder bottoms) and frying a host of foods: vegetables like bell peppers, onions, hash browns; meats like pork chops and bacon; and we mustn't forget grilled cheese sandwiches. Frying pans are good, everyday, useful pans.
If you're a real gourmet, you might want a copper pan lined with stainless steel; copper picks up heat immediately and loses it as soon as the pan loses contact with the burner. But for most of us, stainless steel, perhaps wrapped around an aluminum core, is more practical and economical. You might use a cast iron skillet, which takes special care to maintain but retains heat well and browns food admirably. Non-stick pans can never brown as well as regular pans, but they are convenient, especially for omelettes, and require less fat than regular ones. Be sure with a non-stick pan that you don't cook over high heat or use metal utensils or abrasive scrubbers, as all these will damage the coating.