Overview
Clans are a feature added in Update 53 that let you and other players team up together and make a Clan which allows you to fight for a high ranking spot on the leaderboards with other Clans. The leaderboards track the following stats:
Clan Elo (Elo of every player in the clan combined)
Clan Gold (Gold of every player in the clan combined)
Clan Gems (Gems of every player in the clan combined)
Clan Creation
To create a Clan you first have to make sure you aren't in any other Clan. Once you've checked this, click on the Clans ui which looks like this: 
After you open the Clans menu, click the "Create Clan" button: 
Requirements
The requirements for creating a clan are:
Minimum 80 player level
5000 Gems
If you have both of those you can start creating a Clan. While creating a Clan you can make a Clan Name consisting of 20 characters, Clan Tag which can be any 3 letters you choose (keep in mind that both Name and Tag are filtered by Roblox automatically), and add an Image which can be anything. You just have to upload the Decal or Image ID.
Here is what the Clan creation menu looks like: 
If you have done everything correctly you can start inviting players and begin to compete with other Clans! The Clan limit for players is 100 so you don't have to worry about running out of space fast.
Clan Ranks
As of now there are only 3 ranks in Clans:
Owner - Obtained only after making a Clan yourself. Allows you to invite other players to your Clan, edit the Clan's name, tag and image (the first edit will be free but after that each edit will cost 100 gems), gives you permission to kick any player from the Clan, lets you promote other members to the Officer rank and lets you Send a Mail to your entire Clan for 200 gems.
Officer - Obtained by getting promoted by the Clan Owner. Allows you to invite players to the Clan and kick players with the Member rank in your clan.
Member - Rank you get by default after joining a Clan. You don't get any special benefits from this rank.