what is siege?

Is this a variant of chess or a special type of tournament?

No, it isn't a variant.
Think of it as " king of the mountain" with a chess board.
The winner holds the board and defends against each challenger.
Opponents wait in line to challenge the current board holder, one at a time.
If you win the game you become the defender until someone beats you.

If you click the 'sieges' link there is an explanation on the left side of that page.
I think it's subscribers only, however.