Can I get information about the covenant in-game in Dark Souls?

There's no way of finding out all of the information for the covenants in-game. Some things are hinted, others you will find through trial and error. For in-depth information regarding covenants I advice you to check one of the several Dark Souls wikis, such as these: Dark Souls Wikidot, Dark Souls Wikispaces or Dark Souls IGN Wiki Guide.

There are nine covenants in the game.

As a general rule, joining a Covenant and following it's rules provides you with one or several rewards. Sometimes, there's also a penalty for betraying a Covenant (killing a NPC aligned with such Covenant or joining a different one).

Some covenants allow you to rank up your alignment by offering a particular item, in order to get additional rewards.

Here's some info I gathered from my first play-through and some info digging. (This list is not complete):

Way of White

How To Join

• Talk to Petrus of Thorolund in Firelink Shrine
• You can also join by speaking to Rhea of Thorolund after rescuing from the Tomb of Giants. She will be praying inside the church at Undead Parish after her rescue

Rewards

• Grants immediate access to Miracles sold by Petrus of Thorolund:
  • Heal
  • Great Heal Excerpt
  • Homeward
  • Force
  • Seek Guidance

• Grants higher priority to be matched with online players of these Covenants:
  • Way of White
  • Warrior of Sunlight
  • Princess's Guard

Penalties

• Lose one Estus Flask

Notes

• You can still get Petrus' miracles later on in the game without having to join this covenant

Blade of the Darkmoon

How To Join

• Find the Darkmoon Seance Ring in The Catacombs. If you equip it, a hidden path, located where the second bonfire in Anor Londo lies will be revealed. You will have the option to interact with the carpet before the fog wall and join the covenant

Rewards

• Awards the Blue Eye Orb which allows you to invade the worlds of players who have sinned
• Awards the Blade of the Darkmoon Covenant Ring which, when equipped, summons you to worlds of players who have killed Gwynevere and turned Anor Londo dark
• Awards the Darkmoon Blade Miracle (Rank 1)
• Awards the Darkmoon Talisman (Rank 1)

Notes

• Crossing the fog gate next to the carpet in Anor Londo will prevent you from joining the covenant for the rest of the game
• Killing players as a Blue Phantom awards you with Souvenirs of Reprisal which can be used to rank up this covenant
• Ranking up this covenant beyond Rank 1 (up to Rank 3) strengthens your Darkmoon Blade Miracle

Princess's Guard

How to Join

• After defeating Ornstein and Smough in Anor Londo, you'll have access to the chamber where Gwynevere is. Speak to her to join this covenant

Rewards

• Awards the Ring of the Sun Princess
• Grants higher priority to be matched with online players of these Covenants:
  • Way of White
  • Warrior of Sunlight
  • Princess's Guard

• Grants "permission" to use the following Miracles:
  • Soothing Sunlight
  • Bountiful Sunlight

Notes

• You can still obtain the Soothing and Bountiful Sunlight miracles for the trophy, you just can't use them without being a member of the covenant

Path of the Dragon

How To Join

• Talk to the Everlasting Dragon in Ash Lake

Rewards

• Awards the Everlasting Dragon Eye which allows you to challenge other players online to earn Dragon Scales
• Awards the Dragon Head Stone
• Boosts your Estus Flask x10 without any bonfire kindling required
• Awards the Dragon Torso Stone (Rank 2)

Notes

• You can trade in Dragon Scales to rank up this covenant
• The use of Dragon Stone and Dragon Torso allows you to transform into a dragonoid. In this form you can cast a flame spell
• Ranking up this covenant strengthens your dragonoid flame spell

Forest Hunter

How to Join

• Speak with Alvina, the cat in Darkroot Garden. Answer YES to all of her questions

Rewards

• Awards the Cat Covenant Ring
• Spawns the vendor Shiva of the East
• Awards Divine Blessing consumable (Rank 1)
• Awards the Ring of Fog (Rank 2)

Notes

• When equipping the Cat Covenant Ring, you may be summoned to protect the forest from other players online
• Ranking up this covenant is achieved by slaying intruders. One kill for Rank 1 and three kills for Rank 2

Warrior of Sunlight

How To Join

• Level up your Faith to 25 and head to the bonfire beneath the Red Drake in Undead Parish. Next to it you will find an altar you can pray at, allowing you to join the covenant

Rewards

• Awards the Lightning Spear Miracle
• Awards the Praise the Sun Gesture (You can obtain it without joining)
• Grants higher priority to be matched with online players of these Covenants:
  • Way of White
  • Warrior of Sunlight
  • Princess's Guard

Penalties

• Takes the Lightning Spear Miracle away from you

Notes

• You can substantially lower the Faith requirement by helping online players defeat area bosses. Each boss kill lowers the requirement by 5 points. To be summoned to other worlds, place a summoning sign using the White Soapstone obtained from Knight Solaire

Darkwraith

How To Join

• Talk to Darkstalker Kaathe in the Abyss. To get to the Abyss you will need the key to the seal in New Londo Ruins from Ingward, an NPC in the same area. Also, if you have obtained the Lordvessel, it's mandatory that you do not speak to Frampt (which prompts you to place the Lordvessel on the Firelink Altar).
• You will also have to kill the Four Kings before being able to speak to Kaathe.

Rewards

• Grants you the Dark Hand (Shield/Weapon that allows the absorption of Humanity)
• Grants you the option to buy the Cracked Red Eye Orb (Rank 1)
• Grants you the Dark Armor Set and Dark Sword (Rank 2)

Notes

• Ranking up this covenant is achieved by turning in Humanity.

Gravelord Servant

How To Join

• To join, you have to talk to Gravelord Nito who lies inside an area in The Catacombs, only reachable via an empty coffin, next to a Titanite Demon. Of course, this must be done before you kill Gravelord Nito as part of the story progression.

Rewards

• Awards the Gravelord Sword
• Awards the Gravelord Sword Dance Miracle
• Awards the Gravelord Great Sword Dance Miracle (Rank 1)
• Grants you the ability to earn Souls and Eye of Death for killing cursed players

Notes

• In order to rank up this covenant, you must turn in Eyes of Death dropped by Basilisks or obtained through PVP.

Chaos Servant

How To Join

• Speak with Queelag's Sister in order to join. She lays pass a secret door, on the lower portion of the tower where you ring the second bell, after killing Queelag.

Rewards

• Awards the Great Chaos Fireball Pyromancy
• Awards the Chaos Storm Pyromancy (Rank 2)
• Opens up a shortcut from Demon Ruins to Lost Izalith (Rank 2)

Notes

• Ranking up this covenant is achieved by turning in Humanity.