There are four types of material from which a frame can be built:
Bicycles are traditionally made from steel, a simple alloy of iron with 1/3% carbon. It is easy to work, strong and forgiving. Huge numbers of inexpensive bicycles are still made in steel. Lighter bicycles can be made from stronger alloys of steel eg a mixture of Chromium and Molybdenum ('Cr-Mo'), along with nickel, manganese and tungsten (something like 95% iron and 4.75% other). Further weight savings can be made by using butted tube, thinner in the middle of straight portions, thicker at the joints (where the bending stresses are greatest and welding has upset the heat-treatment a little). Steel welds well and doesn't much suffer from fatigue, so the weight of it can be designed right down. This gives a slightly flexible, responsive and comfortable frame. The thinnest tubing is not much more than egg-shell thick and dents easily. Steel (or "alloy") tubing is relatively easy to weld and repair and generally fails gradually, giving the rider some warning. Steel requires good paintwork and internal treatment to prevent rust.
Aluminium alloy (or "ali") frames can be designed lighter than steel, but only with care, since they cannot not be allowed to flex. Ali fatigues more quickly than steel, so it will likely crack eventually, whereupon it will fail suddenly. It is more or less impossible to repair. It is protected by an oxidisation process ("anodised") and usually covered with a clear laquer.
Carbon fibre produces an extremely light and strong (for the weight) frame, so is often used on top end racing bikes. The main disadvantage is that carbon is very expensive as a material and it needs very careful handling during manufacture.
Titanium is another alternative to steel. It is light weight (lighter than aluminium), strong, shock absorbing and it does not rust ever. Disadvantages are that the raw material is relatively expensive and more difficult to fabricate than steel or aluminium (due to its inherant strength and high melting point).
What you often see, is a bike with an aluminium frame and a carbon fork (for shock absorption). This combination appears ideal for those, who are not cash rich. This combination is relatively cheap, absords shock, does not rust easily and it is a strong bike!