What is the advantage in using copper cookware?

copper is an excellent conductor of heat. My favorite are lined with teflon. I have quite a few lined with tin and you have to watch out because tin melt at a low heat. Copper is excellent for long cooking things as you can generally lower the temperature a bit and the copped is thicker than most people's normal cookware which helps prevent scorching the bottoms and turning whatever you are cooking into something bitter.

If you count bowls, a copper bowl (no lining) is great for whipping egg whites (the copper interacts with the eggs, no need for cream of tartar).

Copper is an excellent conductor of heat and also loses the heat easily. This helps the chefs to have more control over the contents inside the utensil, enough heavy to sit on the burner securely but is also enough light to be lifted easily. And very beauty
Detail:
http://bestcoppercookware.net/choose-best-copper-cookware/

This is way more detail than you asked, but:

http://forums.egullet.org/topic/25717-understanding-stovetop-cookware/

I bought a copper sautee pan after reading this article and have been happier with it than any other cookware I own.

If you make a lot of sauces, a copper Windsor pan (narrower bottom, flares wider at the top) or a saucier are very useful. Tin or stainless lined, of course. The wider top creates more surface area for evaporation. There's a ton of information on the Internet about people's preferences. You could try googling a phrase such as, "what is the difference between copper and stainless." And yes, does require a lot of upkeep but it is beautiful.

People use copper, because it conducts heat so well (no hot spots). And of course, it's beautiful.

Copper's downsides are that it's reactive (so for most applications, its lined with tin or stainless steel), it requires maintenance, and it's expensive.