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.

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

Covenants

• Aldrich Faithful
• Blades of the Darkmoon
• Chaos Servants
• Dragon Remnants
• Mound Makers
• Pilgrims of Dark
• Rosaria's Fingers
• Spears of the Church
• Thieves' Pact
• Vinheim Scholars
• Warriors of Sunlight
• Watchdogs of Farron
• Way of White

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 1.69.