There are several readings you could do to educate yourself about the important events that took place on this site.

Quote from Apollodorus: “In attempting to violate Alcippe, Halirrhothius, son of Poseidon and a nymph Euryte, was detected and killed by Ares. Impeached by Poseidon, Ares was tried in the Areopagus before the twelve gods, and was acquitted.”

Quote from a speech by Demosthenes: “First, then, in ancient times, as we are told by tradition, in this court alone the gods condescended both to render and to demand satisfaction for homicide, and to sit in judgement upon contending litigants,–Poseidon, according to the legend, deigning to demand justice from Ares on behalf of his son Halirrothius, and the twelve gods to adjudicate between the Eumenides and Orestes. These are ancient stories; let us pass to a later date. This is the only tribunal which no despot, no oligarchy, no democracy, has ever dared to deprive of its jurisdiction in cases of murder, all men agreeing that in such cases no jurisprudence of their own devising could be more effective than that which has been devised in this court. In addition to these great merits, here, and here alone, no convicted defendant and no defeated prosecutor has ever made good any complaint against the justice of the verdict given.”     

Another quote from Demosthenes: “You are all of course aware that in the Areopagus, where the law both permits and enjoins the trial of homicide, first, every man who brings accusation of such a crime must make oath by invoking destruction upon himself, his kindred, and his household; secondly, that he must not treat this oath as an ordinary oath, but as one which no man swears for any other purpose; for he stands over the entrails of a boar, a ram, and a bull, and they must have been slaughtered by the necessary officers and on the days appointed, so that in respect both of the time and of the functionaries every requirement of solemnity has been satisfied. Even then the person who has sworn this tremendous oath does not gain immediate credence; and if any falsehood is brought home to him, he will carry away with him to his children and his kindred the stain of perjury—but gain nothing. If, on the other hand, he is believed to be laying a just charge, and if he proves the accused guilty of murder, even then he has no power over the convicted criminal; only the laws and the appointed officers have power over the man for punishment.”

Quote from Pausanius: There is also the Hill of Ares, so named because Ares was the first to be tried here; my narrative has already told that he killed Halirrhothius, and what were his grounds for this act. Afterwards, they say, Orestes was tried for killing his mother, and there is an altar to Athena Areia (Warlike), which he dedicated on being acquitted. The unhewn stones on which stand the defendants and the prosecutors, they call the stone of Outrage and the stone of Ruthlessness.

Read The Eumenides of Aeschylus which tells the story of the trials (and tribulations) of Orestes, son of Agamemnon, who killed his mother and her consort to avenge the murder of his father. His trial for matricide took place on the Areopagus, according to legend.

http://www.amazon.com/Oresteia-Agamemnon-Libation-Eumenides-Classics/dp/0140443339/ref=sr_1_1?ie=UTF8&s=books&qid=1249421626&sr=8-1

Check out this very informative article:
http://www.stoa.org/projects/demos/article_areopagus?page=all&greekEncoding=unicodeC

Read also the account of the Apostle Paul’s address to the Areopagus Council: The New Testament, Acts 17:16-34.