describing a movement
When you sit or sit down, you lower your body until your bottom is resting on something. The past tense and -ed participle of sit is sat.
You usually use sit rather than ‘sit down’ when you mention the place where someone sits.
A woman came and sat next to her.
Sit on this chair, please.
If you are not mentioning the place, use sit down.
She sat down and poured herself a cup of tea.
saying where someone is
If you are sitting somewhere, your bottom is resting on something such as a chair. In standard English, don't say that someone ‘is sat’ somewhere.
They are sitting at their desks.
She was sitting on the edge of the bed.