We'll start with the basics here. At their most basic, missiles are either launched from the surface (including the sea) or the air. Knowing that, deconstructing basic terms is pretty simple. An SAM is a surface-to-air missile, like the Patriot system, while an AAM is an air-to-air missile, like the US military's AIM-9 Sidewinder (fun fact, the "AIM" designator is short for air-intercept missile and is common on AAMs used by America's armed forces). There are, naturally, also ASMs, or air-to-surface missiles, such as the AGM-65 Maverick or the drone favorite, the AGM-114 Hellfire. Surface-to-surface missiles, meanwhile, are common artillery weapons.
Like the ASM (also known as air-to-ground missiles, or AGMs), anti-radiation missiles are launched towards surface targets by aircraft. They are specifically designed, though, to eliminate the electronic emissions from a SAM system. US forces are partial to the AGM-88 HARM, or high-speed anti-radiation missile.
The big, scary missiles, of course, are well known. Ballistic missiles, which in their scariest configuration carry the ICBM, or intercontinental ballistic missile, designation, are the main category of a family of missiles identified largely by their total range. Alongside ICBMs, there are short-, medium- and intermediate-range ballistic missiles.
Perhaps the most recent evolution of missile technology is as anti-satellite weapons, which can consist of either surface- or air-launched missiles designed to target objects in space. These weapons are known to cause severe problems for not just the targeted satellite, but satellites in the same general vicinity or orbital path, as they're known to create a huge, potentially damaging debris field. China was the subject of international outrage after one of its anti-satellite missiles struck a defunct orbiter in 2007, putting a number of other satellites at risk.