Regular aluminum pans aren't suitable for foods with much acid in them -- tomato sauces, vinegars, etc. This is because the aluminum reacts with the acid, which both alters the taste of the food and harms the pan itself. Otherwise, aluminum is inexpensive, heats very evenly, and is light enough to handle easily. Aluminum pans warp. Lightly made aluminum pans dent very easily.
Aniodic Aluminum, a.k.a. Hard Anodized Aluminum:
Some manufacturers, using electrolysis, bond a dark grey or black aniodic layer to the exterior and interior surfaces of the pans. This makes the surfaces non-reactive. It does not make them non-stick. Some people think these surfaces can be "seasoned" to make then non-stick, but they are wrong. These pans behave in exactly the same way as any aluminum pans, with all of their virtues and vices -- except they are not reactive. Two manufacturers, Calphalon and Lincoln (commercial sources only), really got it right. A third, All-Clad makes a beautiful aniodic aluminum but uses it as the exterior only for pans with a stainless interior.
Stainless steel pans are non-reactive and are suitable (chemically at least) for all types of foods. Stainless steel is available in a number of grades and hardnesses. Some grades are more stainless than others. Some steels are so soft they scratch very easily. Good stainless is comparatively expensive, comparatively heavy, warps easily and does not heat evenly.
Better, modern stainless pans are not entirely stainless. They either have a fat aluminum or composite diffusion disk to prevent on the bottom to prevent hot spots; or are made in layers of various materials including aluminum which extends all the way up the sides, with a stainless insert forming the inside of the pan and contacting the food. Thiese constructions resolve most of stainless' problems, including warpage, but if well-made these pans also tend to be heavy.
For most people, the advantages are so great, they dominate the middle and better home cookware market. This degree of market penetration means there are a lot of manufacturers competing for customers. Most are good, but many are charlatans and scam artists who should be avoided. If you're getting ready to equip your first kitchen, this is probably what you should be looking for as your "starter set."
Non-stick pans are, by and large, made of some other material -- usually aluminum -- which is coated with some type of slippery plastic. The plastic coating cleans very easily. In addition, some foods may be cooked with less oil or other fat, giving the illusion that that non-stick cooking is "healthier."
However, most of the coatings are made from materials that scratch and chip relatively easily, then transfer some of the plastic to the food. In addition, the construction prevents foods from searing properly and developing something called "fond." Fond are the crystallized juices formed by high heat and proteins or sugars which stick to the bottom of a pan, and after a "deglaze" are used to make pan sauces. In addition, most non-stick pans require the cook to use special tools to prevent scratching.
After all these years, non-stick pans are still controversial in better restaurant kitchens and among professional and other good cooks. Non-stick pans are certainly not all created equal; there are certainly some which are much better than the run of the mill. Most home cooks find a piece or two of non-stick to be useful. I don't own any non-stick, but understand the best brand for quality/value is Swiss Diamond.
Copper is sometimes used as one of the "plys" is multi-ply bottoms or multi-ply pots and pans. While copper is one of the more conductive materials, there's not enough copper in these sandwiches to make a difference. Its presence is pure marketing, meant to trick you into wanting an expensive distinction without a performance difference.
Many pots and pans with copper exteriors are built to similar specification, and although significantly more expensive (and beautiful) will not cook better than their aluminum or stainless clad counterparts. There are several manufacturers selling this sort of "copper" cookware, notably All-Clad. Unless you're buying for looks, avoid it. Some copper pots and pans, usually from Europe are made with thick copper exteriors (at least 1.5mm), and stainless inserts. Here, the copper actually conveys a slight performance advantage as opposed to an aluminum exterior, or aluminum sandwiched between layers of stainless. However, the advantage is slight and is well beyond the point of "diminishing returns." Still, if you want the very best, love the looks and can afford it...
50 or 60 years ago, the "best" cookware was copper coated with tin. However, tin coatings do not hold up well to abuse. It used to be easy to get copper retinned, but now it's difficult, time consuming and expensive. Don't buy. While uncoated copper can be used for a few purposes -- beating egg whites by hand for instance -- it's generally unsuitable for cooking because it's so highly reactive. Don't buy. Revere and a few other manufacturers coat the bottoms of some of their lines with a copper wash. Absolutely worthless.
There are a variety of other materials used for cookware. The most common so far unmentioned are the various types of steel, cast-iron, enamel over steel, and enamel over cast-iron. Each of these as its virtues and vices. However, the sort of discussion that would convey enough information to be useful would fill the chapter of a book. I'm not willing to go deeply into it, at least right now.
What's best for you?
This is one of those "depends" questions. Mostly it depends on your budget and what you already have. For most people, on a budget, starting without much of anything, it's best to buy a small, mid-priced set of multi-ply stainless, or at least stainless bonded to a diffusion disk; then slowly add a variety of pieces made of whatever materials are most appropriate for their purposes and aesthetics. In my experience people with professional cooking backgrounds tend to be more eclectic and have more fun squeezing a dollar, while people without tend to like more matching pieces and are willing to spend a bit extra. Truth is at a decent level of quality -- other than for highly specialized pieces, a paellera for instance -- it really doesn't make much difference. People obsess.
I get the feeling there's an underlying questions here. You want to know what kind of cookware you should want and buy. The more you tell us about yourself, what you think you'll be doing with your pots and pans, and how much you're willing to spend, the better we can help.