The Third Age began after the first downfall of Sauron, when he was defeated by the Last Alliance of Elves and Men following the downfall of Númenor.
This age was characterized by the waning of the Elves, the rise and decline of the exiled Númenorean kingdoms, and the slow recovery of Sauron's power.
It lasted for 3021 years, until Sauron was again defeated, this time finally, when his Ruling Ring was destroyed. When Bilbo and Frodo left Middle-earth for the Uttermost West, the Fourth Age began.
Compare Babylon 5's opening announcement: "It was the dawn of the Third Age of Mankind".