It was the religious, athletic, artistic, political, judicial, social, and commercial meeting place for the ancient Hellenes: the agora (Ἀγορά), or 'gathering place'. As many ancient Hellenic cities were planned from the envisioned agora outwards, the agora was often located either in the middle of the city or near the harbour, which was surrounded by public buildings and by temples. In many cities, and for most of the age of Hellas, the agora was the centre--the heart--and it served a multitude of functions.

http://www.mlahanas.de/Greeks/Maps.htm

It seems that '[i]n the 5th and 4th centuries BC two kinds of agora existed. Pausanias, writing in the 2nd century AD, calls one type archaic and the other Ionic. He mentions the agora of Elis (built after 470 BC) as an example of the archaic type, in which colonnades and other buildings were not coordinated; the general impression created was one of disorder. The agora of Athens was rebuilt to this type of design after the Persian Wars (490–449 BC). The Ionic type was more symmetrical, often combining colonnades to form either three sides of a rectangle or a regular square; Miletus, Priene, and Magnesia ad Maeandrum, cities in Asia Minor, provide early examples. This type prevailed and was further developed in Hellenistic and Roman times.'

Very early on in Hellenic civilization, the agora became the focal point of Hellenic life. In the 7th and 6th centuries BC, the agora was not only the central meeting place for the living, but a honourable burial place for the very important dead. It was a practice that slowly faded as death became to be viewed as more and more miasmic, but very old graves have been found at locations of agora's.

The location was always more for the living, however, and tended to be the hub of social activity. The agora was designed to be easily accessible to every citizen, and often housed a large central square for market stalls, which were ringed by a colonnade, sometimes containing shops, or stoae. Statues, altars, trees, and fountains adorned the square itself, and it was bound by public buildings.

The agoras of the ancient Hellenic world varied greatly in size, but in gneral, the size of the agora was in line with the size of the city. The agora in Athens, for example, was the size of several football fields and saw heavy traffic every single dag. Women didn't often frequent the agora, but it was the central meeting place for anyone else, from politicians, philosophers, traders, city officials and slaves to the city's criminals.

In many cities, the agora would have been crowded with confectioners who made pastries and sweets, slave-traders, fishmongers, vintners, cloth merchants, shoe-makers, dress makers, and jewelry purveyors. Men and slaves usually did the shopping, with slaves and donkeys carrying the purchases; wealthier women may have visited to buy perfumes, jewellery and expensive cloth. In Athens, a special separate 'potters market' was reserved for the buying and selling of cookware as that was considered solely the provenance of women and was frequented by female slaves on task for their mistresses or by the poorer wives and daughters of Athens.

Kapeloi, retail traders, were a frequent sight at the agora, and served as middle-men between the craftsmen and the consumer, but even then, these vendors were deemed unnecessary and were often mistrusted by the general public.

Scientific theory got its start in the agora, where the city's greatest minds regularly met informally to socialize. In Athens, Socrates, Plato and Aristotle were frequently seen at the agora, discussing philosophy and instructed pupils. Hippocrates and Pythagoras were both highly public figures who taught and shared ideas in their own hometown agoras. Democracy found its start at the agora as well.

As a religious site, the agora makes an odd but somewhat understandable choice; many temples were focussed around the agora, and in all centuries, the agora was lined with altars to the Gods--including special altars to the dodecatheon, the Twelve, the major gods of the city--and many statues of Gods, heroes, and those who had earned their statue to be erected at the square. Olympic victors often earned that right, as well as those who had become war heroes or those who had served the city in another important capacity.

Athena and Zeus in Athens especially had special epithets connected to the agora: Agoraios and Agoraia, patrons of the agora itself. the agora further served as both a meeting place, a sacrificial area, and the main area for public celebrations where meat from the sacrifice was distributed amongst the city's citizens. The agora--while rarely the focal point of worship--tended to be at the start and end of religious ceremonies, and would have been the designated area for discussions about the rites and many of the activities connected to it.

The concept of  city centre is mostly lost to us now, although some small towns and yes, some larger cities especially those not in the Western world, still make use of a large open space where all ways of life interconnect, and where the main focus of social interaction is placed. With its central location and open character, the agora was the logical place to spend free time, and because religion was such a large part of the ancient Hellenic way of life, it's only logical that religious activities also featured heavily at the agora.