Roman chariot racing owes much to the Greek tradition which preceded it. Literary references to Greek-chariot racing can be found in several sources, such as Homer (The Iliad, Book 23, lines 262-565), who describes the event in great detail, and Sophocles (Electra, lines 45-55), for example.

The Greek chariot race involved a track with a left-handed turn around a tree or post back to the starting line, a format maintained by the Romans. This format has striking similarity to modern horse racing as well as car racing (NASCAR, in particular), and was adopted by the Romans (probably from either Greek or Etruscan sources) as early as the fifth century B.C.E.

By the later Republican period, the chariot racing hippodrome had evolved from a rectangular arena surrounding two posts at opposite ends to the more familiar circus with a low wall called a spina between the two posts. This wall was primarily to prevent head-on collisions, though the rules did not disallow attempts by charioteers to drive their competitors into the wall itself (just as modern racing drivers may attempt to force their competitors off the track), which occasionally resulted in spectacular collisions (also seen in modern car racing).

The teams you refer two, the Greens and Blues, were popular in the later Empire period, though chariot racing in Rome existed long before the Empire. Tertullian claims in De Spectaculis that the original teams were the Reds and the Whites (De Spectaculis, Book One. "On Public Games," Section IX). Eventually the Greens and Blues were also formed, and by the late empire period, the Reds and Whites had merged into them.

A chariot racing team could have up to three competitors (leading teams of between two and ten horses, depending on the era) in this format; chariot races did not always consist of multiple teammates, nor were these four teams the only competitors involved in chariot racing.

There is little record of chariot racing in the early Republican period, though Elizabeth Rawson [speculates] (http://www.jstor.org.ezproxy2.library.arizona.edu/stable/40310870.) that wealthy private citizens may have purchased the equipment and horses to participate in the games of the period. Later, slaves were the primary competitors.

These slaves were trained in chariot racing, and when they won races were considered the winners (as opposed to their masters) and could claim prizes for themselves. Eventually, some of these slaves were able to purchase their freedom with their winnings. Most famously, the charioteer Diocles received prizes totaling approximately 36 million sesterces (an enormous sum in gold, making him the highest-earning sportsman of all time).

It was during Nero's reign that chariot racing came to be especially popular with emperors. Nero himself participated in chariot racing, including at the Olympics, where Suetonius records that he drove a "ten horse team" and was awarded the crown despite crashing and failing to finish (Lives of the Twelve Caesars. Nero, 24.).

The racing teams became more popular during the later Empire period, especially in Byzantium, and by the reign of Justinian they had achieved immense wealth and even political power. Violent brawls occured between supporters of the Greens and Blues, and historians like Evans and Hathaway note that the emperor himself usually supported one of the factions.

Basically, Roman chariot races consisted of multiple teams of horses, pulling a chariot crewed by a single man which travelled around an oval track divided in the middle by a low wall connecting two posts. In the Greek period, the race consisted of a dozen laps, but by the later Empire period this had been reduced to seven.

There are many parallels to modern racing - the shape of the track is nearly identical to modern NASCAR tracks, for example. You mentioned being unaware of teams in racing, but there are actually modern racing teams. The format which is most similar to Roman chariot racing in this regard is Formula One, where multiple drivers from the same time (i.e. Ferrari, McLaren, etc.) compete in the same race, but the individual driver is crowned the victor.

Sources:

C. Dodgson, Tertullian, De Spectaculis. Vol. 1. Apologetic and Practical Treatises, pp.187-219. Trans. 1842.

Harris, H. A. "The Starting-Gate for Chariots at Olympia." Greece & Rome 15, no. 2 (1968): 113-26.

Rawson, Elizabeth. "Chariot-Racing in the Roman Republic." Papers of the British School at Rome 49 (1981): 1-16.

Schrodt, Barbara. "Sports of the Byzantine Empire." Journal of Sport History 8, no. 3 (1981): 40-59.

Suetonius. Lives of the Twelve Caesars, "Life of Nero." Loeb Classical Library. Trans. 1914.