So, this varies wildly by companion, but I'll try to run down all of them.
Quest: Ed-E My Love
Trigger: Recruit Ed-E, then speak to Johnson Nash in Primm, either in his shop (after installing a new Sheriff), or at the Vicki and Vance (when you first arrive in town).
Reward: If you send Ed-E to the Brotherhood of Steel, he'll receive an upgrade to his armor (+8DT), if you send him to the Followers, they'll a-charge up his lazors (~+50% damage, exact amount unconfirmed).
Notes: Ed-E's companion quest require that the robot 'overhear' conversations with certain key NPCs about topics such as the Enclave, Pre-War Technology, etc. There is also an inherent time delay between the triggers - you can not trigger Log 2 until 9 days after Log 1, and you will not hear from Lorenzo or Martimer for 2 days after that.
Quest: Nothin' But a Hound Dog
Trigger: This quest is acquired as you recruit Rex. Just speak to The King about Rex, then Julie Farkas over at the Old Mormon Fort, and then return to The King with the news that Rex can be healed. Rex is now recruited as a companion, and you'll have this quest, to take him to Jacobstown, in your log.
Reward: If you give Rex Violetta's Brain, he'll receive a speed increase. Rey's Brain increases his attack. Lupa's brain increases his max health.
Notes: You can also use an NCR Guard Dogs brain for this quest, but Rex will gain no additional perk if you do so. Lupas brain is currently bugged - it increases Rexes current health by 100, but not his maximum - once he takes 100 damage, the benefit is gone forever.
Quest: Heartache By the Numbers
Trigger: Talk to Cass at the Mojave Outpost. That's it. She won't join you, and you won't be able to begin working on this quest in earnest however, until you've completed the first few objectives of You Can Depend On Me, bought out Cassidy Caravans, and actually recruited Cass as a companion.
Reward: If you turn McLafferty and the Van Graffs in to the authorities with evidence, Cass will gain the Calm Heart perk (+50 HP). If you instead kill both, she'll gain the Hand of Vengeance perk (+15% damage with Guns).
Notes: Calm Heart appears to have the same bug as Lupas brain does for Rex - it increases current, rather than maximum health.
Quests: Crazy, Crazy, Crazy(to Recruit), Old School Ghoul(Unmarked quest to 'upgrade' Raul)
Trigger: To trigger Crazy, Crazy, Crazy, simply go to Black Mountain. Neil will introduce himself and the quest will be added to your log. Once Best Friend Tabitha is dealt with, Raul will be free to join you. Old School Ghoul will never appear in your log. To complete it, you simply need to introduce Raul to Ranger Andy, Corporal Sterling, and Loyal, and talk to Raul about each.
Reward: After completing Old School Ghoul, you can have a conversation with Raul about his role in the world, and his wild 'youth'. If you convince him to return to his gunslinging days, he'll gain the Old Vaquero perk, increasing his attack speed with Revolvers and Lever-Action firearms by 33%. He'll also put on his old Cowboy outfit. If you instead convince him that his mechanical skills are more useful, his Regular Maintenance perk will be upgraded to Full Maintenance - reducing the rate at which equipment decays by 75%. He'll also incorporate some armored padding into his Petro-Chico jumpsuit.
Notes: The triggers for Old School Ghoul can be inconsistent - particularly, if you've ever spoken to Ranger Andy before meeting Raul, the dialog will sometimes not trigger. As of now, the only way known to get around this is to use the console on the PC. See the wiki for full details.
Quest: I Forgot to Remember to Forget
Trigger: Boone probably has the toughest Companion Quest to trigger. In order to get him to open up to you about what happened at Bitter Springs and begin this quest, you need to gain 5 'trust points' with Boone. The easiest way to gain these points is killing Legion with Boone as a companion. Lots and lots of Legion. Specifically, you can gain:
Reward: Boones default armor upgrades to 1st Recon Survival or 1st Recon Assault Armor (depending on conversation choices). The difference is purely cosmetic - both offer a DT of 15.
Quest: I Could Make You Care
Trigger: As with Boone, Veronicas quest is triggered by the accumulation of 'trust points'. In Veronicas case, these are gained by taking her to a variety of locations in the Wasteland and talking to certain NPCs. You need to have 3 of these conversations to trigger Veronica's Quest:
Reward: If you convince Veronica to stay with the Brotherhood, she'll gain the Bonds of Steel perk (+4 DT). If you instead convince her that she's better off leaving to join the Followers, she'll gain the Causeless Rebel perk, which increases her unarmed attack speed by 30%.
Notes: There is also an unmarked 'side quest' for Veronica. She wants a dress. More than anything in the world, she just wants a really pretty dress. Give her some Formal Wear (easiest to find at the Ultra-Luxe), and she will be overwhelmed with gratitude - she'll teach you the Scribe Counter, a special attack usable with Unarmed weapons.
Quests: Guess Who I Saw Today, Unnamed additional 'quest'.
Trigger: For Guess Who I Saw Today, just talk to Doctor Henry about the Nightkin Schizophrenia and his search for a cure. This makes Lily a recruitable companion. In addition, once the Courier has travelled with Lily for a bit, you may begin to hear a recording of her grandchildren. Once you've listened to the whole recording, you can discuss it with Lily.
Reward: After discussing Lily's grandchildren, you can give her advice about how she should continue with her medication. If you recommend she stop taking it, she will gain some permanent stat increases, but she will continue to frenzy and become uncontrollable at half health. If you suggest she continue to take half-doses, she will only frenzy at 25% HP instead of 50%, and will calm down immediately if you sneak. If you suggest she switch to full doses, she will never frenzy, but will suffer some permanent stat penalties.
Quest: For Auld Lang Syne
Trigger: As with Boone and Veronica, Arcades quest can be triggered through a point system - however, you can completely skip this process if you are idolized with the Followers of the Apocalypse. To gain points with Arcade:
- Complete side quest That Lucky Old Sun with Arcade, and direct the power from Helios One to Freeside.
- Take Arcade for a visit to the Crashed Vertibird, the REPCONN Test Site, REPCONN Headquarters, the Silver Rush in Freeside, or the Nuclear Test House at the Old Nuclear Test Site.
- Complete side quest The White Wash - framing the Scorpion gang is Arcades preferred outcome, but even if you arrest Anderson, so long as you keep Westside supplied with Water, he'll be happy.
- Finally, you can bypass this whole process and jump right to the end by taking Arcade to The Fort after receiving the Mark of Caesar. When he objects to being taken right to the heart of the Legion, tell him that you're just there to talk, see what Caesar is up to, and get out as quickly as possible.
Additionally, Arcades quest WILL NOT TRIGGER until you have locked yourself in to one of the four end-game paths(excluding Caesar's Legion, which will not trigger the quest). If you have not yet locked yourself in, the dialog necessary to trigger this quest will not appear.
Rewards: Power Armor Training, Remnants Power Armor (arguably the best armor in the game), support from the Remnants in the final battle (If Legion/NCR. If you're running Wild Card or with Mr. House, you may end up fighting them.) With a speech check, you can also get the Gannon Family Tesla Armor.
Notes: After completing this quest, you will permanently lose Arcade as a companion. It is unclear however, whether this loss is intended, or a bug.