(Spoilers)
List your order of your favorite companions. And tell why if you want. Only list the one's you've actually made a companion before.

Mine:
-Raul Tejada:
I love his Revolver and he can repair my stuff whenever I need him to.
-Arcade Israel Gannon:
He's awesome. And if you do his quest you can get one of the best armours in the game. Enclave Power Armour. The only reason he's not my favorite is because when you go against the NCR he wants to leave you. And does after you keep hurting NCR.
-Rex:
He's a cute little dog. What's not to love.
-Craig Boone:
He's an awesome sniper who has a very great shot. And he's a BA. Only reason he's not up higher on my list is the same as Arcade, fight the NCR he'll leave you.
-Veronica Santangelo:
She's pretty great. And uses one of the best Melee weapons I've ever come across.
-ED-E:
I love this little robot. But it's hard to "connect" with a companion that isn't actually alive.
-Rose of Sharon Cassidy (Cass):
She's pretty cool. Is helpful around the wasteland. But not my favorite.
-Lily Bowen:
Honestly the only reason she's not up the list is I can not take a giant Nightkin in a dress and guardening hat seriously.

The Companions:
http://fc01.deviantart.net/fs71/i/2010/309/8/f/fallout__new_vegas_companions_by_doomed_dreamer-d31n6l8.png

Here's is a little about each Companion:
-Arcade Israel Gannon, a member of the Followers of the Apocalypse in 2281 who can be found in Freeside's Old Mormon Fort.

• -Craig Boone, a former NCR 1st Recon sniper with a grudge against Caesar's Legion and a chip on his shoulder. He can be found at night in the dinosaur at Novac. Can be recruited after completing One for My Baby and using the right dialogue.
  
• -Lily Bowen, a friendly nightkin and former assassin in the Master's army. She can be recruited in Jacobstown after you get the quest Guess Who I Saw Today.
  
• -Raul Tejada, a ghoul mechanic and former gunslinger held hostage by Tabitha, the mad leader of the State of Utobitha. Can be found in Black Mountain.
  
• -Rose of Sharon Cassidy (also known as Cass), a former caravan merchant and the daughter of John Cassidy. She can be found in Mojave Outpost.
  
• -Veronica Santangelo, a disillusioned Brotherhood Scribe. She can be found at the 188 Trading Post.

-ED-E, a prototype eyebot designed for combat and reconnaissance operations. It can be found in Primm, in a non functional state, lying on the counter inside the Mojave Express Which is labeled in-game as "Nash Residence". ED-E requires 65 Repair or higher to fix him and 3 pieces of scrap Metal, 2 sensor modules, and some scrap electronics.
-Rex, is a cyberdog in Freeside under the ownership of the King. He is suffering from neural degradation as his old brain decays. He can be recruited after finishing Nothin' But a Hound Dog.
(Wikipedia information)

Arcade Gannon. Love him for his witticisms and light-hearted atttitude. His sardonic sense of humour is right up my alley. I also think he has one of the most interesting backstories; it must be hard living not knowing who you really are. I can sympathise there. Just kit him out with the Q-35 and you're set. Of course, being an Enclave supporter, I'm slightly biased

Like Gannon, Raul is one of the funniest characters in the game; his random quips are brilliant. That [censored] Vaquero outfit is about the only thing I dislike about Raul - a bit too ridiculous.

Veronica's good, too, but some of her one-liners feel a bit... forced, like she's always trying to make a joke. Still well written, though the same goes for all of the companions really.

Boone's a badass, but a total drag, and athough I can understand his plight, I cannot imagine how it could ever be possible to hang around a guy like that.

ED-E is basically the canon follower of the Courier and with the upgrades from the Divide, there ain't [censored] getting between me and ED-E Again, anything Enclave-related piques my interest.

I don't really care for the others, especially Lily. Cass and Rex are cool; maybe it's me. I haven't really found a way to make them effective.

On a side note, I don't think Arcade leaves you if you oppose the NCR. So long as you don't side with the Legion he'll stick around. He prefers you work toward the independence of New Vegas, but it's not essential.

My favorite companion is Marcus. He wasn't a companion in NV but he was in Fallout 2. Therefor he should count.
Favorite New Vegas companion is Ed-E.