Clan challenges are set up between clan leaders. There can be any number of people in the challenge - 1 player vs 1 player, or 20 vs 20 - and there's only one "round". Who plays who is negotiated between the clan leaders. Once one of the two clans wins, that clan gets points. The clans are listed by these points on the Clans list; for example, my clan, the Freethinkers, is number 3 on that list as we've won the third greatest number of clan challenges of all clans.
The Leagues are set up by the site administration I think. There are several Leagues, each with a different time control. Each clan can submit exactly six people for their team in each League ranked in a specific order, usually highest rating to lowest rating to start. For example, I am fourth board on the Freethinkers team in the 3/7 League because when the League started I was the fourth highest of the six people we were submitting.
Opponents for each player are based on some system I am not aware of. This system seems roughly to be based on people of similar ranking within their respective teams playing one another but it's not exact apparently; despite my being fourth board, I am playing MATIC, who is fifth board for his team.