In this video the Sun is at the center, with the inner planets (and occasionally Jupiter) seen rotating around it from a position above the plane of the solar system (so looking down on the Earth's North Pole). The planets aren't labeled, but it's easy to count outwards from the Sun to find Earth. At the start of the video in 1970 all of the 4,422 asteroids known at that point are marked with tiny dots. (It's best to watch on the highest resolution you can, otherwise they're hard to see.) As time unfolds, asteroid discoveries are marked with bright white dots, before fading to a fainter color. Yellow and red dots (I'm guessing) are asteroids that come close to Earth, or Near-Earth Objects (NEOs).
Later in the video you might notice asteroids in the same orbit as Jupiter. These are known as the Trojan asteroids. These asteroids orbit the Sun in roughly the same orbit as Jupiter, but 60° either ahead or behind of it. The name comes from the fact that early on the first few discovered were given names from the Trojan War (starting with 588 Achilles), and it was proposed to continue with the naming scheme and give them all such names. The naming scheme even extends to location: asteroids ahead of Jupiter are given names from the Greek side of the war, while those trailing it are given names from the Trojan side. (Though amusingly, there are two out-of-place names from before this particular convention was adopted: Patroclus is found among the Trojans, and Hektor among the Greeks.)
The term Trojan asteroid originally referred to just the asteroids in the orbit of Jupiter, but when other asteroids sharing similar orbits with other planets were found the term expanded to encompass them as well. Currently Trojan asteroids are known for Earth (1), Mars (7), Jupiter (6,000+), Uranus (1), and Neptune (13).
As for why Trojan asteroids are generally found around 60° in front of or behind the planets they share an orbit with, that has to do with gravity, the three-body problem, and Lagrange points, and really deserves a blog post to itself sometime.
And finally, while watching these nifty visualizations, just keep in mind the fact that sizes are not to scale. In reality, while it looks like the asteroid belt is a swirling maelstrom of space debris, there are typically millions of miles (or kilometers!) between any two asteroids. There may be a lot of rocks out there, but there's an even larger volume of space to hold them. A hui hou!