Copper pans are loved for being quick to warm and for their extremely even distribution of heat. They are often specifically called for in candy-making and other heat-sensitive recipes, but copper is useful for almost all your cooking, since the increased control and evenness of temperature makes everything easier.
Although copper is one of the most expensive materials for pans, for those who love to cook copper pots and pans are worth the investment. They also aren't difficult to take care of, despite what many believe. Like cast iron, the key is to understand the material, why it works, and what its weaknesses are.
Copper conducts heat supremely well because it is a soft metal. Copper as a metal is highly reactive and thus isn't foodsafe on its own, but lined with a non-reactive metal — nickel, tin (older pans), or stainless steel (newer pans) — it's entirely safe to cook in. When buying used copper pans, always inspect the interior of the pan to make sure this lining is intact, with no pitting and no verdigris (blue copper patina). Almost all of the care with copper pans is about maintaining this lining.
Copper is so conductive people often have to relearn what temperatures to cook different foods at. A good rule of thumb is to use half the heat (or flame) you would on a non-copper pan, and never ever heat an empty pan. For example, if in a stainless steel skillet you cook eggs on medium-high, use medium low for copper. The reason for this is two-fold: to not burn the food, and to not melt the tin lining (tin is also a soft metal which melts around 450°F).
Because of this temperature restriction, I wouldn't recommend using copper pans in an oven above 325°F, since most ovens fluctuate 50°F or more in both directions while maintaining a single temperature. Additionally, watch out for extremely acidic or salty foods, as long-term contact will corrode your pan.
Whether or not you choose to polish your copper is really a personal preference. To polish it, you can use special copper polish or anything lightly acidic — lemon, vinegar, or tomato juice work particularly well. Just make sure to clean the pan with soap and water after. If you don't polish it, the tarnish will actually protect the copper underneath from wear. Don't worry about polishing the inside; tin naturally darkens with use, and there is little you can do to prevent it.
If you have hard, baked-on foods such as caramel, it's better to soak your copper pans rather than scrub them. Always use the soft side of the sponge to avoid scratching the soft, thin layers of metal.
Most of all, enjoy your copper pans. Try out a soufflé (which also isn't as hard as reputed!), make pancakes on low heat and experiment with how fast they cook, and feel like Julia Child as you admire your pans on the wall.