At its most simple, steel is iron with carbon in it. Other alloys are added to make the steel perform differently. Here are the important steel alloys in alphabetical order, and some sample steels that contain those alloys:
Carbon: Present in all steels, it is the most important hardening element. Also increases the strength of the steel but, added in isolation, decreases toughness. We usually want knife-grade steel to have >.5% carbon, which makes it "high-carbon" steel.
Chromium: Added for wear resistance, hardenability, and (most importantly) for corrosion resistance. A steel with at least 13% chromium is typically deemed "stainless" steel, though another definition says the steel must have at least 11.5% *free* chromium (as opposed to being tied up in carbides) to be considered "stainless". Despite the name, all steel can rust if not maintained properly. Adding chromium in high amounts decreases toughness. Chromium is a carbide-former, which is why it increases wear resistance.
Manganese: An important element, manganese aids the grain structure, and contributes to hardenability. Also strength & wear resistance. Improves the steel (e.g., deoxidizes) during the steel's manufacturing (hot working and rolling). Present in most cutlery steel except for A2, L-6, and CPM 420V.
Molybdenum: A carbide former, prevents brittleness & maintains the steel's strength at high temperatures. Present in many steels, and air-hardening steels (e.g., A2, ATS-34) always have 1% or more molybdenum -- molybdenum is what gives those steels the ability to harden in air.
Nickel: Adds toughness. Present in L-6 and AUS-6 and AUS-8. Nickel is widely believed to play a role in corrosion resistance as well, but this is probably incorrect.
Phosphorus: Present in small amounts in most steels, phosphorus is a essentially a contaminant which reduces toughness.
Silicon: Contributes to strength. Like manganese, it makes the steel more sound while it's being manufactured.
Sulfur: Typically not desirable in cutlery steel, sulfur increases machinability but decreases toughness.
Tungsten: A carbide former, it increases wear resistance. When combined properly with chromium or molybdenum, tungsten will make the steel to be a high-speed steel. The high-speed steel M2 has a high amount of tungsten. The strongest carbide former behind vanadium.
Vanadium: Contributes to wear resistance and hardenability, and as a carbide former (in fact, vanadium carbides are the hardest carbides) it contribute to wear resistance. It also refines the grain of the steel, which contributes to toughness and allows the blade to take a very sharp edge. A number of steels have vanadium, but M2, Vascowear, and CPM T440V and 420V (in order of increasing amounts) have high amounts of vanadium. BG-42's biggest difference with ATS-34 is the addition of vanadium.