What a sonotube is for
A 'Sonotube' (the brand that became the generic word) is a heavy cardboard cylinder set in the ground as a form, so a deck footing or column comes out round, smooth and plumb. Typical deck footings are 10–12 inch tubes sized to the beam load, poured down to the frost line with a footing pad or flared base at the bottom.
Size the tube to the load, not the hole
The tube diameter follows the load: a typical residential deck footing for one post is 10–12 inches; a ground-level porch post may only need 8. Your local deck code table (based on tributary area and soil bearing) is the authority — the tube should also be a couple inches wider than the post base hardware so anchors have room.
Pouring tips
Wet the tube inside so it does not steal water from the mix, pour in lifts and rod each lift to push out air, and set the anchor bolt while green. Leave the tube on through curing — stripping it too early in freeze country exposes green concrete.