This is a pretty complicated subject, and the various wikis have a lot more detailed information about them. In general, you are ranked from +0 to +3 in your current covenant; some covenants give items or other benefits for attaining ranks, and joining some covenants will let you use covenant-specific items and miracles. They're pretty complicated (no two are alike, really, beyond the fact that you can only ever be in one at a given time), but here's a summary of the stuff you can get:
- Way of White: No items or ranks, but it does seem to make co-op easier (you see more White Soapstone signs as a human)
- Warrior of Sunlight: gain the use of the Lightning Spear miracle for joining, Greater Lightning Spear miracle at rank 2, seekrit spoiler miracle in NG+. Makes co-op easier, and gives you a golden White Soapstone sign (like Solaire of Astora). Rank up by delivering Sunlight Medals to the Sunlight Altar; you can gain the medals by defeating bosses in co-op (you don't have to be in the covenant to get them). Oh and you also get a gesture for joining (Praise the Sun!)
- Princess's Guard: No ranks, but you get the Ring of the Sun Princess and the Soothing and Bountiful Sunlight miracles. Makes co-op easier, too. Note that unlike most covenant miracles, you don't actually need to be in the Princess's Guard to use those two, and you can get them from other places.
- Path of the Dragon: Gain the Everlasting Dragon Eye and Dragon Head Stone for joining, along with an upgraded Estus flask - you always get at least 10 charges, even if you're resting at a bonfire that has not been kindled. The Dragon Head Stone gives you a dragon's head when you use it, and lets you use a fire breath attack. Every rank in the covenant upgrades the fire breath attack. You also gain the Dragon Stone Torso at rank 2. This covenant is mostly for dueling, as the Everlasting Dragon Eye lets you challenge other members of the covenant to a fight; the winner (or both, if a tie) receives a Dragon Scale.
- Gravelord Servant: Gain the Gravelord's Sword item and the Gravelord's Sword Dance miracle for joining, as well as the ability to use Eyes of Death (and generate more of them, if you use them properly). Gain the Gravelord's Greatsword Dance at rank 2.
- Forest Hunter: Gain the Cat Covenant Ring and access to a merchant for joining; the NPCs in the forest behind the Crest-Locked door won't attack you any more (you can't talk to them, and attacking them breaks the covenant). Gain a (single, consumable) Divine Blessing at rank 2, and the Ring of Fog at rank 3. Wearing the Cat Covenant Ring will cause you to occasionally be summoned to the forest, in order to defend it against players. Defeating these players will improve your rank in the covenant. Note that breaking this covenant is pretty much the only way to gain the Dark Wood Grain Ring (which lets you do flippin' sweet backflips), as the only one in the game is carried by the merchant's bodyguard.
- Blade of the Darkmoon: Gain the Blue Eye Orb and the Darkmoon Covenant Ring for joining. The Blue Eye Orb lets you invade and attack players who have "sinned", and wearing the Darkmoon Covenant Ring will occasionally summon you to attack players who have done a bad thing, much like the Cat Covenant Ring. Successfully killing the target of either of these will grant you a Souvenir of Reprisal, which you use to rank up in the covenant. At rank 1 you gain the Darkmoon Talisman and the Darkmoon Blade.
- Darkwraith: Gain the Dark Hand for joining, which lets you steal humanity from other players and NPCs. Gain the Red Eye Orb at rank 1 (an unlimited use version of the Cracked Red Eye Orb), and the Dark Sword + Dark Armor set at rank 2.
- Chaos Servant: Gain the Great Chaos Fireball pyromancy for joining, and the Chaos Storm pyromancy + a shortcut to Lost Izalith at rank 2 (note that this shortcut is the only covenant-specific area).