The process in this condemnation was thus : the people being assembled, every man took a tile called ostrakon and carried it to a certain part of the market place, surrounded by wooden rails for that purpose, in which were ten gates for the ten tribes to enter distinctly : in this place the tiles were deposited by each person, and numbered in gross by the archons.
Source: wiktionary
Also on view are ostraca, pottery fragments on which Athenians inscribed the names of persons they felt too powerful for the good of the city and deserving of ostracism, or 10 years’ exile, a procedure formalized by Cleisthenes.
Source: wiktionary
If there was sufficient interest, the Athenians could vote once per year to ostracize someone, exiling him from Athens for a period of 10 years. To vote, Athenians scratched the name of the person to be ostracized on potsherds, broken pottery pieces called ostraka—hence the name of the procedure.
Source: wiktionary