Why ten? Why not seven or eleven? Rabbinic legend traces this back to the Biblical story in Genesis. God tells Abraham that He is going to destroy the city of Sodom. Abraham haggles, saying, “You are a just God, you would not destroy the innocent along with the guilty, would you spare the city if there were 100 good people?” The haggling gets down to ten – that God would spare the city for the sake of ten decent people. (In the end, there ain’t even ten, so Sodom gets blasted.) Thus, ten is the minimum number of decent people required for survival in the face of evil, and so ten was chosen as the minimum number for communal prayer.
Throughout the Bible, ten is a number of completeness – Ten Commandments, ten generations between Adam and Noah, ten generations between Noah and Abraham, ten plagues on Egypt, etc.
OK, so we’ve already had the comments on community prayer compared to individual prayer.
A few additional comments. The presence of ten creates communal prayer, required (for example) for ceremonial reading of the Torah, saying the memorial prayer, etc. There are special prayers added for a group (for example, a lengthy silent devotion is repeated aloud by the leader.) Fewer than ten, it’s individuals praying together; ten or more, it’s a community praying.
The role of prayer in traditional Judaism is different from its role in traditional Christianity. In traditional Judaism, prayer replaced sacrifice when the Temple was destroyed; prayer is therefore an obligation, part of the requirement of being called God’s first-born (Chosen), rather than a pouring out of the soul. Much of prayer in Judaism is therefore ritualized rather than internalized.
Final comment. Other ancient peoples were conquered and disappeared. Judaism survived, survived the destruction of its center (the Temple in Jerusalem). One reason for that survival was the earlier invention of community prayer, in local synagogues, that helped bind the people together (since you needed to find ten), even when scatterred around the globe. Thus, Jews tended to live together (even if not forced to do so), which undoubtedly helped to preserve the traditions and the people-hood.