Managers and scorers
A tournament has managers, who can do everything that can be done to it, and may have scorers, who enter its scores and nothing else. Whoever created the tournament is its first manager.
Managers
The Managers tab lists who runs the tournament, in the order they were made, with your own name marked as yours. It is where a tournament is shared, or handed on.

Somebody is made a manager by their email address. Type it into the field above the list and press Add Manager. The address has to belong to an account that is registered and confirmed; an address with none is refused beneath the field, and the site says no more about it than that. A scorer made a manager stops being a scorer.
Remove takes a manager off. Nothing is kept of a removed manager, so Remove asks first; they are made a manager again by their address if they are wanted back. A tournament always has a manager, so the last one is not offered Remove.

You can take yourself off. The question then says what that means: you lose your own way back to the tournament, and nobody on the page can undo it for you. Another manager can make you one again.
Scorers
The Scorers tab is the Managers tab for scorers, and works the same way: an email address and Add Scorer above the list, and Remove on each.

A scorer sees the tournament in their own list in the admin, with Scores in place of Manage, and is offered its tests alone: no teams, notes, managers, scorers or rankings. They cannot open a team's page, since who plays for a team is the managers' to say. On a test's page they can enter, correct, archive, restore and reorder scores as a manager can.

A manager is not made a scorer, since a manager can already do everything a scorer can. A tournament needs no scorer, so every scorer can be removed, the last among them.
Next: Rankings.