Plato's dialogues, and to a lesser extent Aristotle's writings, have conveyed to later generations an impression that the Academy was the physical location of the 'golden age' of Greek philosophy. If this notion is perhaps misleading, it is none the less influential. University teachers are all nowadays 'academics'. 'Academe' as something of a literary conceit is used by Shakespeare in Love's Labour Lost.
Physically, the Academy was a garden open to the public, six stadia outside the walls of Athens, by the side of the river Cephissus. It is supposed to have contained a sacred grove of olive trees dedicated to Athena (the goddess of wisdom). Plato is thought to have acquired property there around 387 BC and to have taught there, together with other philosophers and aristocrats.
The name seems to have come from its being part of an estate said to have belonged to Academus. The inspiration behind the Academy however, was the school established by Pythagoras at Croton. When Plato died, leadership of the Academy was passed to his nephew Speusippus, who was later followed by Sceptics such as Arcesilaus and Carneades. With the fall of Athens in AD 88, all the buildings were destroyed. 
Heads of 'The Academy'
Plato's immediate successors as Head (or Diadochus) of the Academy were Speusippus (347-339 BC - and hence not Aristotle), followed by Xenocrates (339-314 BC), Polemon (314-269 BC), Crates (ca. 269-266 BC), and Arcesilaus (ca. 266-240 BC).
Later heads include Lacydes of Cyrene, Carneades, Clitomachus, and Philo of Larissa, considered to be the 'last undisputed head of the Academy'.
- see, for example, http://www.britannica.com/EBchecked/topic/2615/Academy
- 'Essentials of Philosophy and Ethics', edited by Martin Cohen, Hodder Arnold 2006
- The Cambridge History of Hellenistic Philosophy (Cambridge University Press, 1999), pp. 53-54