What is a minyan?

The source

The Torah says “I will be sanctified in the midst of the Children of Israel” (Leviticus 22:32). The Talmud links this, through the word “midst” and the word “congregation” (edah) used of the ten spies (Numbers 14:27), to a group of ten: “any matter of holiness requires no fewer than ten.”

What requires a minyan

Kaddish, Barchu, Kedushah, the reader’s repetition of the Amidah, the priestly blessing, the public Torah and Haftarah readings, and the seven blessings at a wedding. One may pray alone, but the prayer of a community is especially accepted.

Counting

There is a custom not to count people by number; instead a verse of ten words is used (“Hoshia et amecha…”). Orthodox practice counts men aged thirteen and over; egalitarian communities count women equally.

When there is no minyan

Someone praying alone says the Amidah and the Shema, but omits Kaddish, Kedushah and Barchu; ideally he prays at the same time as the local congregation. The Shulchan Aruch says a person should make an effort to pray with a community, and in Jewish tradition someone who comes to complete a minyan is doing a great mitzvah.

Mourners and the minyan

Mourners rely on the minyan to say Kaddish. That is why during shiva friends gather in the house of mourning for the daily prayers, and why many synagogues run a daily minyan even with small numbers.

Ten, even if not all pray

Someone who is still in the middle of his own Amidah is counted towards the ten for Kaddish and Kedushah. A boy below bar mitzvah is not counted in common practice, Ashkenazi or Sephardi.