The minyan is the quorum of ten required for communal prayer, and its logic is that certain sacred acts belong to the community rather than to the individual and cannot occur without it.
What requires ten
A minyan of ten is needed for the parts of the service that are held to require the presence of the congregation: the repetition of the standing prayer, the reading of the Torah, the responsive sanctifications, and, most keenly felt, the mourner Kaddish, which a bereaved person can recite only where ten are gathered. A person praying alone says a reduced service, omitting exactly those elements, so the difference the minyan makes is precise and visible.
Who counts
In Orthodox practice the ten are adult men, which ties the ability of a mourner to say Kaddish to the availability of a male quorum and has made the counting of women a defining division within Judaism: Conservative, Reform and Reconstructionist communities count women in the minyan, and Orthodoxy does not. The requirement shapes Jewish settlement in a concrete way, since a community must be large enough to guarantee ten for daily prayer, and the effort to secure a minyan, especially for a mourner, is one of the ordinary labours of Jewish communal life.