This theater of Dionysos was built in the 4th century B.C. to replace and enlarge the earlier theater in which the plays of the great Athenian dramatists were first performed. The new theater seated some 17,000 spectators in 64 rows of seats, 20 of which survive. Most spectators sat on limestone seats -- and probably envied the 67 grandees who got to sit in the front row on thronelike seats of handsome Pentelic marble. The most elegant throne belonged to the priest of Dionysos (god of wine, revels, and theater); carved satyrs and bunches of grapes appropriately ornament the priest's throne.
Herodes Atticus, a wealthy 2nd-century-A.D. philhellene, built the Odeion -- also known as the Odeum or Irodio (Music Hall). It is one of an astonishing number of monuments funded by him. If you think it looks well preserved, you're right: It was reconstructed in the 19th century. Although your 2€ entrance ticket for the Theater of Dionysos allows you entrance to the Odeion, this is misleading. The Odeion is open only for performances. The best ways to see the Odeion are by looking down from the Acropolis or, better yet, by attending one of the performances staged here during the Athens Festival each summer. If you do this, bring a cushion: Marble seats are as hard as you'd expect, and the cushions provided are lousy.