The source is in Talmud, Brachos 21b and Megilah 23b. See also Ramba'm,
Sefer Ahava, Hil. Tefila ch.8. In short, the Talmud derives the rule "Kol
davar shebekedusha lo ye'hei pachos me'asara" ("All holy acts must be
with no less than ten") from the verse "ve'nikdashti b'soch bnei yisrael"
("I shall be sanctified among bnei yisrael") (Vayikra 22:32), through a
rather lengthy process, to establish that this pasuk is referring to an
"eidah" ("assembly" is the most common translation) which the Talmud shows
to mean a minimum of ten.
There are additional benefits from tefilah b'tzibor (prayer with a
minyan) which while not always absolute requirements are still
considerable. To quote the Ramba'm, Sefer Ahava, Hil. Tefila ch.8:1, "The
prayers of a congregation are always accepted. Even when there are
sinners among them HaShem is never repelled (or disgusted) by the
prayers of a multitude. Therefore it is necessary for a man to join a
congregation and not pray alone whenever it is possible for him to pray with
a congregation.... Anyone who has a synagogue within his city and does
not pray in it with the congregation is considered a bad neighbor."
Kesivah ve'chasima tova!
Lazer