Covenants in Shadowlands
When you enter the Shadowlands on your first character for the first time, you will be required to follow the entire storyline. You'll start in Bastion and progress through Maldraxxus, Ardenweald, and finally Revendreth. When you have completed this story arc and have reached level 60, you will be given the choice to align yourself with one of the four Covenants. You must align with one of them for the expansion's storylines to progress further.
All subsequent characters you take into the Shadowlands will have the option of choosing a Covenant right from level 50. You aren't required to go through the entire storyline again for every character. You may choose to do so, but you can just jump right into a Covenant from the beginning, start working on world quests and jump right into end-game progression.
Choosing your Covenant is based on many factors, including your class, desired class Covenant ability, playstyle, aesthetics, and roleplay sensibilities. There may be many people telling you what is "best" for your class, but what is best for one purpose (mythic+, raiding) may not be best for another (solo questing, PvP, or roleplay). It needs to be an individual decision based on many factors, and one you need to make carefully. Switching Covenants is easy; switching back to one you'd previously joined and left is another matter.
To progress through the systems of your chosen Covenant, you will need to earn Renown. This Covenant-specific metered, level-tiered system measures your progress of earning favor with your chosen Covenant.
Each Covenant will have three Covenant members and creatures with whom you can form a Soulbind. When you first join a Covenant, you will have access to two of these Soulbind characters, and will have an opportunity to unlock a third through an epic Covenant quest. It doesn't make that denizen a bodyguard, like the followers in Warlords of Draenor or Legion. It's more akin to having the perks available through the artifact weapon system, without anything you need to carry around and that isn't tied to a particular spec.
You can use Calculator for Covenant and Soulbind to choose one.