Entering any covenant found throughout Lothric affects the multiplayer mechanics in various ways.
Completing duties for a covenant grants the player certain tokens, which can be returned to the covenant leader to unlock unique rewards. Each covenant also grants a certain bonus while equipped.

Covenants can be switched on the fly by equipping their respective covenant item.

Covenants

Covenant	Leader	Token	Effect while equipped
Aldrich Faithful	Archdeacon McDonnell	Human Dregs	Boosts Dark damage by 5%
Darkmoon Faithful	Company Captain Yorshka	Proof of a Concord Kept	Restores 1% FP every second
Blue Sentinels	Company Captain Yorshka	Proof of a Concord Kept	Increases max FP by 5%
Chaos Servants	Cornyx of the Great Swamp	Fire Seed	Boosts Fire damage by 5%
Company of Champions	Domhnall of Zena	Awestone	Boosts Souls gained from defeated enemies by 10% Boosts enemy damage and absorption by 30%
Dragon Remnants	Hawkwood the Deserter Archdragon Peak altar	Dragon Eye	Boosts Stamina Damage by 25%
Mound Makers	Sacrificial Altar	Vertebra Shackle	Boosts Physical damage by 5%
Pilgrims of Dark	Karla	Abyssal Fragment	Increases casting speed by 15
Rosaria's Fingers	Rosaria, Mother of Rebirth	Pale Tongue	Boosts Souls gained from defeated enemies by 10%
Spears of the Church	Filianore's Altar	Filianore's Spear Ornament	Restores 3 HP per second
Thieves' Pact	Greirat of the Undead Settlement	Titanite (no offerings)	Boosts Item Discovery by +25
Vinheim Scholars	Orbeck of Vinheim	Crystal Fragment	Boosts Magic damage by 5%
Warriors of Sunlight	Altar of Sunlight	Sunlight Medal	Boosts Lightning damage by 5%
Watchdogs of Farron	Old Wolf of Farron	Wolf's Blood Swordgrass	Reduces weapon Stamina Consumption by 10%
Way of Blue	High Priestess Emma	Various consumables (no offerings)	Increases max HP by 5%
Way of White	Irina of Carim	Holy Fragment	Boosts Estus Flask recovery by 10%

Notes

• Covenants can be switched to on the fly without penalty. Covenants cannot be switched while playing as a Phantom.
• Your current covenant is displayed in a special slot within your inventory.
• Specific enemies have low chances of dropping your Covenant's Tokens.
• When playing as or with an allied player Phantom, you and your ally gain half the normal drop chance for each other's Covenant's Tokens.
• You must discover a covenant before you can switch to it.
• Multiple copies of covenant rewards can be obtained as long as the player has enough offerings.

Information here is accurate for version 1.79.