Plenty of bands have two guitar players. Plenty of other bands get away with having the lead vocalist play nothing, and only have one guitar. And a few other bands get away with a single lead vocalist/guitar player.
What are the pros/cons of the various configurations?
Some bands like the fuller sound of two guitars. You can have one guitar playing lead lines over the rhythm of the other guitar. It’s really just a matter of taste. The White Stripes have no bass player and they sound just fine too. Led Zeppelin only had the one guitarist for concerts (not counting when John Paul Jones did some guitar or mandolin or whathave you), but on their albums they layered the guitars over and over.
Depends who the guitarists are. Duane Allman and Dickie Betts – or Duane Allman and Eric Clapton – pretty damn good.
Couldn’t have said it better. 
The specific benefit that guitar pairings served in these two bands is that the guitars were able to harmonize, so that adds an entire new dimension to a band’s sound. And when you get two guitarists who are skilled and work well together, then they can play off of each other to great effect. Sometimes one plays rhythm while the other solos, but what makes it really interesting is when there’s a real back-and-forth and they throw ideas at each other.
The main reason for having two guitar players is so you have one to steal the glory from the bass player, while the other steals it from the drummer!
Yeah! Just look at what happened to Iron Maiden! 
Having two guitarists didn’t seem to bother the Beatles much!
As a guitarist who just recently moved from a power trio where I was the singer and guitarist, to a two-guitar format with other singers along with me, the differences are many - and there is no right answer:
With a single guitar - I could improvise more without having to worry about coordinating with the other guitar. Selfishly, I got to play all the cool parts - if that meant a crunching rhythm - that’s me. Cool lead - me again! Since guitarists are often raging ego-types (who - me?) this can be a great advantage.
With two guitars: You can fill in the sound live. There is nothing cooler than playing, say, Aerosmith’s Walk This Way, where I am the one guy singing/rapping out the lines, and playing Joe Perry’s syncopated riffs underneath the verses (try it sometime it is HARD - I had a ton of guitarists come up to me asking me how I did it) - but there is nothing WORSE than transition to the lead bits and having no rhythm guitar to fill in the sound - it just empties out the sound and loses momentum. That is why “true” 3-piece bands like Cream or the Who or Rush tend to feature bassists who play a form of “lead bass” - they climb into the tonal range of a rhythm guitar. Zep and Van Halen both uses multi-tracked guitars heavily when in the studio.
I would also say that a 3-piece can lead to a looser feel. When singing and playing difficult stuff - Walk This Way, Message in a Bottle (!) - I am really doing tough stuff. By just focusing on a guitar part, say, I can deliver that part with much more precision.
I guess that’s it. Sure, as other posters mentioned - you can certainly do two-guitar harmony work with two guitars - and it can sound great. I would also point out the Stones - where Keith Richards’ vision of having two guitars that weave into each is just brilliant - neither is quite fully rhythm or lead and it can be damn near impossible to tell where one leaves off and the other begins.
Even though they sucked, Black Oak Arkansas had a neat 3-guitar (plus bass) configuration. One played 12-string rhythm guitar while the other two did harmony leads, usually in thirds. It’s a nice Southern rock sound.
Without a keyboard, a 2nd guitar is great for filling in harmonies. Unless your last name is Page or Lifeson, you’d have a hard time soloing while filling up those missing chords.
Because two guitars are louder than one.
No glory can ever be stolen from Iron Maiden’s bass player (Steve Harris!). He wrote all the good songs. Bruce Dickenson & company cower to his phenomenalness.
I spent my entire college career trying to get over that hump. Being that I played mostly jazz, I had the advantage of playing a lot of “chord leads.” This is nice because you can reharmonize a passage on the spot. It’s a lot more creative than being locked into the existing harmony.
Plus, when you’re soloing over Giant Steps, no one can tell that you’re not making the changes!
To be precise, it’s one louder.
I know that. I was being sarcastic 
I’ll second everything WordMan said. I was the lead singer / bass player in a three-piece band, and there were many times when I wished we had a second guitarist. We performed a lot of songs where, in the original version, the bass line was very simple, but the rhythm guitar part was a distinct riff. A couple examples are Ted Nugent’s Cat Scratch Fever and Motley Crue’s Looks That Kill. During the guitar solos on those songs, I was forced to figure out how to play the rhythm guitar riff without losing the steady, driving “dumdumdumdumdumdum” of the bass line. I couldn’t just play the straight bass line, because the rhythm guitar riff was essential to the song. At the same time, I couldn’t just play the rhythm guitar part on my bass because then the song would lose the drive and groove of the bass line. I managed, though
Thanks. What inspired this question was Green Day. They’ve always been a trio, but lately they’ve used another uncredited guitarist (or two) when they play live.
Mostly, to cover up the fact that one of the guitarists is drunk/stoned/a pretty boy who can’t play.
BTW–did you hear that the Metal Band, Exodus, has a new drummer? Paul Bostaph.
The song that immediately came to mind after reading the title was Green Grass And High Tides by the Outlaws. Nice blending of guitar work there from at least two, if not three, guitars.
Check out the guitar duet at the end of Hotel California by the Eagles. Then tell me what the point is of having two guitarists.
Again - if you are into harmony leads - a la Hotel California, Southern Rock, etc. - then the benefits of two guitars are clear. Basically, they are essential.
But SmackFu’s question was more about why does a 3-piece band occasionally leverage a second guitarist, especially in live situations?
In Green Day’s case, American Idiot is much more multi-tracked than other GD CD’s, so that matters more in live situations. Same with In Utero for Nirvana, when they brought in Pat Smear as a second guitarist…
[Incredulous]
Who in the world would name their poor child Pat Smear? Don’t answer that… I’m just feeling sorry for the guy.
[/Incredulous]