Definition: The Clone Wars are first mentioned in Episode IV: A New Hope. In her message to Obi-Wan, Princess Leia says, "Years ago, you served my father in the Clone Wars." From Luke's reaction, this was a well-known conflict; but until the Prequels, it was never fully explored. Episode II: Attack of the Clones established that the Clone Wars were fought between the Separatists, a group of plants who wanted to leave the Republic, and the Republic, represented by an army of clone soldiers and Jedi commanders.
In-universe: The Clone Wars lasted from 22 to 19 BBY. The conflict began when several planetary governments joined the Confederacy of Independent Systems, also known as the Separatists. This movement, led by former Jedi Count Dooku, saw the interplanetary Republic government as hopelessly corrupt and sought to secede.
Unbeknownst to the Republic or the Jedi Order, Jedi Master Sifo-Dyas had foreseen a terrible conflict more than a decade earlier and commissioned a clone army from the cloners on Kamino. Obi-Wan discovered this army when he was tracking the bounty hunter Jango Fett, who served as the genetic model for the clones.
The appearance of the army seemed mysterious but miraculous, and gave the Republic what they needed to fight the Separatists. Chancellor Palpatine had no official power to use the clone army, however, and so Jar Jar Binks called for a vote to give the Chancellor emergency dictatorial powers.
The first official battle of the Clone Wars was the Battle of Geonosis, which occurred after Obi-Wan Kenobi, Anakin Skywalker and Padmé Amidala were captured by the Separatists. Mace Windu led a Jedi strike team to rescue them, and later Yoda led the newly-formed clone army into their first combat. The war ended with the Battle of Coruscant, when Chancellor Palpatine was captured by General Grievous. Skywalker rescued Palpatine and killed Count Dooku.
As it turns out, the entire war was a ruse. Palpatine was really a Sith, Darth Sidious, and Count Dooku was his apprentice, Darth Tyranus. Dooku had killed Sifo-Dyas and finished creating the clone army while convincing planets to secede from the Republic. Palpatine used this chaos to justify becoming a dictator -- gaining powers he never intended to give up.