Covenants are "factions" within Dark Souls and Dark Souls Remastered. To join a covenant the player must make oaths to specific NPCs. With each covenant there are rewards for following the guidelines set by your "leader" and there are penalties for breaking them. Covenants may also have an effect on the player's online interactions. There are nine covenants total.
Most covenants have a unique multiplayer item useable only by members of that covenant, and even the ones that do not will have items or spells that can affect other players in some way, and as such what covenant the player has joined will have an effect on the player's online experience. Generally speaking, each covenant will focus on either cooperation (summoning/being summoned) or competition (invading/being invaded), but even then how the player chooses to play the game is still their choice. As an example, a player in the Warriors of Sunlight covenant (which emphasizes co-op pve) could be in the covenant purely for the ability to cast the Sunlight Spear miracle, and use Cracked Red Eye Orbs (an invasion item available to all players regardless of covenant) so that they may invade other players and use the miracle on them!
"The setup for the various covenants affects multiplayer situations in many different ways, especially the parameters of versus play, expanding the scope well beyond Demon's Souls. The game is designed to introduce multiplayer events as an extension of each character's role-playing in these covenants." 
Note: for the achievements you don't need to join them, once you get the dialogue that let you choose whether you want to join, the achievement will pop up.
Covenants of Dark Souls
Mechanics Common to all Covenants
There are nine covenants in the game, each with different goals and different means of achieving those goals, all through multiplayer interaction. While most of the covenants have at least one exclusive multiplayer item for use to this end, there are certain multiplayer items that can be used regardless of covenant: the White Sign Soapstone, the Red Sign Soapstone, Cracked Red Eye Orbs, and the Dried Finger. In this way, any player regardless of covenant can be summoned by another player or invade them, or even if they are in no covenant at all. However, these items will only ever award humanity, and only two out of the nine covenants accept humanity as offerings.
Most of the covenants allow progression within the covenant in the form of ranks, starting at base (referred to as +0), and maxing at +3.. Raising your rank in a covenant will typically give boni related to the equipment or spells associated with that covenant, for example a high-ranking member of the Dark Moon covenant will have a more powerful Darkmoon Blade spell, while a high-ranking Darkwraith might be able to steal up to 10 humanity with a single use of their Dark Hand. New ranks are attained at:
- Base/+0: 0 offerings
- Covenant rank +1: 10 total offerings
- Covenant rank +2: 20 more offerings, for 30 total
- Covenant rank +3: 50 more offerings, for 80 total
In addition, it is possible to donate a maximum of 100 offerings to a covenant, but no further benefits are accrued after 80. The exception to this is the Chaos Servant covenant, in which it is possible to donate up to 60,000 humanity. However, this gives no special boni to your character and will only affect your standing in the Chaos Roster.
Breaking/Abandoning a Covenant
It is possible to leave or abandon a covenant by joining another covenant. However, abandoning a covenant in this manner will count as a sin, which will make the player the target of Dark Moon invasions unless they absolve their sins via Oswald of Carim, for a fee equal to their level times 500. If the player wants to abandon their covenant without this occurring, they must do so by speaking to Oswald.
Furthermore, some covenants have rules or restrictions that, if disobeyed, will "break" the covenant. If this happens, the covenant is immediately abandoned, and the player must absolve their sins through Oswald or else risk the wrath of the Dark Moon covenant.
Regardless of how the player leaves their covenant, half of the offerings given to that covenant are lost and their rank is adjusted accordingly. For example, if the player had given 40 offerings and were at rank +2 before they abandoned their covenant, if they return later their rank will be +1 and they will need to offer 10 more to get back to rank +2. Covenant offerings and rank cannot go below zero in this way.
Dark Souls Remastered: You may abandon a covenant through the bonfire menu. This does not add Sin.
If a player has been invaded and is killed, that player may indict the invader. Doing so will add the invader's name to the book of the guilty, after which they will be subject to invasions by Darkmoon covenant members. The book of the guilty only records the number of times any player has been indicted, and does not count broken covenant vows. Indictments cannot be absolved by Oswald. Oswald will also sell a Book of the Guilty to the player, and by using it the player can see how many times they have been indicted as well as every other indicted player.
It is possible to indict players who have invaded through:
It is not possible to indict a player who has invaded with: