A scheduled thread is the simple version: set a time, and it fires exactly once, using nothing but that thread's own history. Good for "pick this up again tomorrow morning."
An Agent is the version that keeps going. Assign one to a workspace and it re-runs on its own interval indefinitely — this is what people usually mean by a "watcher" or "background" agent. See Tuning how often an agent checks in for adjusting how often.
Each Agent reacts differently depending on how you set it up: some only wake up for a new message, some keep nudging a conversation along regardless, and some have no thread at all and simply go looking for work on their own schedule.
Not all of it has to be a prompt, either. The parts of a loop that are purely mechanical can be written as a script that runs on the same interval with no model behind it at all — see Agents that run code, not a prompt — and a short script can also sit in front of a prompt and decide which threads are even worth waking the model for, which is Letting code decide when an agent runs.
When one Agent hands work to another, it works the way a handoff between two people would — leaving a clear note in a thread the next Agent is already watching, rather than something invisible passed silently between them. A saved, reusable version of this shape is a Strategy. When an Agent needs you rather than another Agent, it can come and find you. If something ever looks stuck, see Reading the Audit Trail.