Eggs are surprisingly tricky to master. While it's easy to claim dominance over one specific style of egg preparation, the sheer variety of egg dishes means that it takes a lot of effort to conquer every way of preparing an egg. Poaching is very different from scrambling, for example, and omelets and over-easy eggs require a totally different touch.
It turns out that the pan you use has a lot of sway over how you prepare eggs. This is pretty different from most other dishes, where a stainless steel pan will perform about as well as a ceramic pan or a Teflon one. With eggs, however, you usually want to be able to slide around your food with the gentlest touch and the least amount of oil. This means that you care about your cooking surface very much.
This also holds true for the material that your pan is made out of. Some people like cast iron for eggs, but for me, it's right out. I need an aluminum or copper core so that my eggs quickly respond to any changes I make in heat level. With a cast iron pan, my eggs will keep cooking long after I turn off the heat. With an aluminum pan, this effect is minimized.
So what kind of a pan is best for cooking eggs? If you're not poaching eggs, you want an aluminum or copper-core non-stick pan with the best non-stick surface you can find. Depending on your preference, this can be a ceramic-based non-stick (like Thermolon) or a PTFE-based non-stick (like Teflon). In either case, you'll want to treat your egg pan with extra care and replace it often, since non-stick tends to get stickier over time.
Both Thermolon and Teflon (and the related family of non-Dupont PTFE non-stick coatings) have their own upsides and downsides. Thermolon requires you to use a bit of cooking oil and supposedly transfers heat a bit better (I've never noticed a difference), but it has the nasty downside of gradually becoming less non-stick over time. This means that you can wake up one morning and fail to make your eggs the way you like them with no warning. On the other hand, it's also free of most of the health concerns that plague PTFE-based non-stick coatings.
PTFE, on the other hand, requires no oil at all. It's also incredibly non-stick and is ostensibly totally safe, depending on who you believe. You should spend some time looking into all of the facts surrounding it on your own. The general idea is that it's pretty safe at room temperature (even if you accidentally eat a few flakes) but it stops being safe if you get it really, really hot. Luckily, you cook eggs well below the temperature at which PTFE becomes toxic.
As far as performance goes, PTFE has the advantage of being predictable. As long as there's PTFE covering your pan, things won't stick to it. Of course, it's vulnerable to scratching or flaking off over time. Both ceramic and PTFE non-stick coatings have a finite useful life of around 2-4 years, depending on how well you care for them.
There's not a huge difference between aluminum and copper pans when cooking eggs. There are definitely egg dishes that require some pretty precise temperature control (like runny French-style scrambled eggs), but you can make all of them just fine in an aluminum pan. Copper makes that easier in theory, but in practice, you'll do just fine with an aluminum pan and a little bit of patience. Just remember: you can't make an omelet without breaking eggs, both literally and metaphorically. You'll definitely "waste" a few eggs on failed attempts as you learn how to cook.
This exact specification of egg pan (non-stick, aluminum, cared for carefully, replaced often) is used by many chefs at restaurants worldwide. While different chefs prefer different types of non-stick and different sizes of pan, the basic formula remains pretty consistent. Once you've got one of these pans, you've got a great tool that you can use to explore the world of eggs.
For more specific information, check out this page at On The Gas. It's got a lot of individual reviews that can help you find the perfect egg pan for your kitchen.