Many ancient Greek democracies had a system by which the dêmos (i.e., the body of adult male citizens) could vote to temporarily banish a certain person from the polis for a period of time. In Athens, this system was known as ostracism and it involved the citizens casting votes by writing the name of the person they wished to have banished on a piece of pottery, known in Greek as an ὄστρακον (óstrakon). The official period of banishment in Athens was ten years.
There were actually many different variations on this system in different Greek poleis. For instance, during the period when Syracuse was a democracy, it apparently had a system known as petalism, which was very similar to ostracism except that people wrote their votes down on olive leaves instead of pottery fragments. (Historians aren't quite sure how this worked, since olive leaves aren't generally very good for writing on.) The Athenian system, though, is by far the most famous and best attested.
Despite the fame of Athenian ostracism, it seems to have only been rarely and sporadically used. In order for an ostracism to occur, the majority of the Athenian assembly first had to vote in favor of holding a separate meeting for an ostracism. Then, according to the Greek biographer and Middle Platonist philosopher Ploutarchos of Chaironeia (lived c. 46 – after c. 119 CE) in his Life of Aristeides 7.5, at least six thousand citizens had to show up to the ostracism meeting and cast ostraka in order for the ostracism to be valid.
If the archons counted any fewer than six thousand ostraka, the ostracism was considered void and no one would be ostracized. If over six thousand total ostraka were cast, then only the person who received the highest number of votes would be banished and they would only be banished for a period of ten years, after which they would be allowed to return.
As a result of these relatively stringent requirements that had to be passed in order for an ostracism to be considered valid, only around thirteen men in total are recorded as having ever been ostracized from Athens, all of whom were prominent politicians and generals.
Additionally, many of the men who were ostracized from Athens were allowed to return to Athens before the period of ten years was over. Xanthippos, Aristeides, and Kimon are all examples of men who were ostracized who were allowed to return to Athens before the full ten years had elapsed. The total number of men who were actually ostracized from Athens for a full ten years is probably less than ten.
The Athenians eventually abandoned the system of ostracism entirely; the last person who is ever recorded to have been ostracized from Athens is the politician Hyperbolos, son of Antiphanes, who was ostracized in around 417/6 BCE. A few years after that, in 411 BCE, there was an oligarchic coup, which resulted in the establishment of the short-lived oligarchy of the Four Hundred. Although democracy was soon restored, there is no record of anyone being ostracized from Athens after this date, which suggests that the system may have been abolished and never re-implemented.
Because the system of ostracism was so seldom used, it probably didn't have an enormous, wide-ranging impact on Athenian democracy at large. Nonetheless, it did have some effects. One effect is that prominent politicians always had to be cognizant of the possibility that the dêmos might ostracize them if they did anything too unpopular.
Another effect is that politicians who were ostracized were no longer available to aid the city if a crisis arose (although, as I noted before, they could be called back if the dêmos felt it necessary and they were willing to return). Finally, a third effect is that the occasional ostracism of a certain prominent politician could allow space for emerging politicians to gain more influence. For instance, the ostracism of Kimon in c. 461 BCE probably helped make room for the rise of Perikles.
For further reading
For information about petalism and other forms of ostracism outside of Athens:
For more about Athenian ostracism and ostracism in general: