Apollo asteroids are Near Earth Asteroids (NEAs) which have their closest approach to the Sun (their perihelion) closer to the Sun than Earth’s orbit, and the furthest distance from the sun (their aphelion) outside Earth’s orbit. This means they cross Earth’s orbit twice every time they make an orbit of the sun. From our point of view, this makes them a particularly risky set of space rocks. There are about 10,000 of these, and all of them are less than 10 kilometres in diameter.
These are an odd group of asteroids, as their comparatively risky orbit-crossing pathways give lots of chances for them to be tugged out of orbit (or, on a very bad day, crash into a planet), and they tend to have expected lifetimes of a mere tens of millions years -- an eye-blink in planetary terms. This means that these asteroids are all newcomers to the neighborhood, dying comets or main belt asteroids that got ejected form their original orbits.
The Apollo group of asteroids are named for their prototype 1862 Apollo -- a nice guy, has his own mini-moon, and while technically listed as a potentially hazardous asteroid, he's probably not going to hit us. It is relevant to note that due to his eccentric orbit, 1862 Apollo is not only an Earth-crossing asteroid, but also a Venus-crossing asteroid and Mars-crossing asteroid. You can see why these guys don't last too long.
There are four classes of asteroid in the Near-Earth Orbit taxonomy. The Apollos are paired with the Atens; both of these groups cross Earth's path two times per orbit, but the Apollos have a semimajor axes larger than Earth's (greater than 1.0 AU, with a perihelion distance of less than 1.017 AU), while the Atens have semimajor axes smaller than Earth's (less than 1.0 AU and their aphelion distance greater than 0.983 AU). The Earth clears a gap between these two groups; any asteroid orbiting with a period of one year (i.e., at approximately 1 AU from the sun), would either be bumped out of orbit through gravitational resonance or captured as an Earth Trojan. Near Earth Asteroids that have orbits smaller than the Earth's (specifically, less than 0.983 AU) are Atira asteroids (this group includes those that orbit much closer to the sun as well, but an asteroid that orbits interior to the orbit of Venus would be more likely termed a Vatira asteroid; interior to Mercury, a Vulcanoid asteroid). An Amor asteroid is one that orbits entirely outside of Earth's orbit, most of them being Mars-crossing asteroids. Technically, the entire asteroid belt consists of Amor asteroids, but don't say that, you'll just piss off the astrophysicists.