After each match, you'll be evaluated based on your performance relative to your team. This affects how many rank points you receive or lose, which in turn results in your promotion/demotion through the various ranks. These ranks affect affect what kind of opponents you're matched against, and can be thought of similar to an MMR system.
Colloquially, we refer to these MMR brackets as bronze, silver, and gold.
You'll also receive a reward each time you reach a new rank. These rewards reset each month and can be earned again after playing a few matches.
Your rank is visible in the pre match lobby as displayed by the colour of the plate behind your name. Pictured here is a silver player surrounded by gold players.
Once you reach 999 points in a given rank, you'll be tasked with playing a series of promotion matches. When promoting through lower ranks, these promotion matches can be as simple as 'Earn at least 1 C rank or higher within your next 3 matches'. Promoting through higher ranks however tasks you with much more strict demands.