Sub-quests

Every act has a handful of optional sub-quests: two or three in a full act, fewer in a half-act. They are short, usually kill a number of monsters in a zone and report back, and they give gold, orbs, a little extra experience and sometimes a special reward.

How to start one. Talk to the NPC after the main quest brings you to town. When you greet them they may offer a quest: say "quest" or "mission". Answer yes or no.

Examples

The Brood Below (Thais): cull the brood and kill the Rotworm Queen for Warden Osric.
Lamps for the Ghostlands (Carlin): thin the dead for Huntress Kyra.
Seeds for the Grove (Ab'Dendriel): clear the roots for Wardruid Aelrin.
Tainted Lamp-Oil (Venore): clear the Tainted Caves for Factor Lucan Vass.
The Collapsed Gallery (Kazordoon): clear the Dwarf Mines for Foreman Durnik.
The Ghostship's Bell (Ankrahmun): a task from Ahmet.
The Watch at Vengoth and The Sunken Quarter (Edron and Yalahar).
Gears and Horns and Floor Inspection (Rathleton).
Echoes of Krailos and The Temple of Equilibrium (Farmine).

There are 17 sub-quests in all across the acts, those of Acts IV to VI included.

Level. Each has a minimum level shown in the quest log.

Optional on purpose. Skipping sub-quests costs you only about a tenth of the experience of the act, never the way forward. They are also a steady source of orbs and gold in the early game.

Progress is tracked server-side per character, so you can leave a quest and resume in another session, even after a death or logout (The quest log).

Repeating. Sub-quests complete once per character.