The writeup above misses out quite a few facts, which I thought noteworthy:
Simon and Garfunkel's recording career actually started in 1957 with the song Hey Schoolgirl, an Everly Brothers soundalike recorded under the pseudonym Tom & Jerry, which apparently had such highly intellectual lyrics as 'oo-boppa=loochy-ba/she's mine she's mine'. The song reached the lower reaches of the charts, aided by an appearance on American Bandstand. However, the follow up singles were unsuccessful, and the duo disbanded, Garfunkel going on to study architecture at university (inspiring Simon's later song So Long, Frank Lloyd Wright) while Simon cut a string of unsuccessful singles under various pseudonyms, and also made a living cutting demos for publishing companies, often with Carole King.
The duo reformed in the early 60s in response to the growing folk scene, and recorded Wednesday Morning, 3AM, their first album, with Tom Wilson, then best-known as Bob Dylan's producer, at the controls. A clean-cut 'hootenanny' style folk album, typical of the period and absolutely unexceptional, featuring versions of Go Tell It On The Mountain, Pretty Peggy-O and simillar folk-club standards, along with a few songs by Simon (the only one of the two who wrote) it was a resounding flop, and the duo split again.
Paul Simon moved over to England, and began touring the folk clubs, where he played with guitarists such as Martin Carthy (from whom he learned Scarborough Fair and probably Davy Graham's instrumental Anji). He also produced an album for Jackson C. Frank at this time; but his main work was an album called The Paul Simon Songbook, a now long out-of print solo album which featured many songs that were later rerecorded by Simon & Garfunkel. The album is out of print as Simon considers it an embarassment.
However, without the duo's permission or knowledge, Tom Wilson had overdubbed electric instruments on a track from their album, The Sounds Of Silence, in the manner of his production of Dylan's Like A Rolling Stone. This was released as a single, and went to number one. Simon quickly moved back to the US, and the duo quickly recorded their second album, The Sounds Of Silence, in the style of the hit, made up mostly of songs Simon had learned or written during his stay in the UK.
The follow-up, Parsley, Sage, Rosemary And Thyme, was a disappointment - sounding rushed, Simon is now thoroughly embarassed by some of the adolescent lyrics-as-poetry on this decent but hardly great album. It did however mark the duo's first collaboration with producer Roy Halee, who would work on all the rest of their albums.
The next album, Bookends, was their masterpiece. Featuring Mrs Robinson (written for the film The Graduate), it featured a new maturity in Simon's songwriting, with the haunting America and enigmatic Save The Life Of My Child
Bridge Over troubled Water, the group's swansong, was almost as good, but songs such as The Only Living Boy In New York and So Long, Frank Lloyd Wright clearly showed the pair were drifting apart, and it was no surprise when they both embarked on solo careers.
The two had reunions occasionally in the 70s, most notably for the single My Little Town (which appeared on solo albums by both), but didn't properly reunite until 1981, for the Concert In Central Park. This was supposed to be the start of a full reunion, and the duo toured and planned to make an album together, but the plans were scuppered when Simon decided his lyrics were too personal to give to Garfunkel and replaced Garfunkel's vocals on the subsequent album, Hearts And Bones with his own.
The two have very occasionally performed together since, and one track was released in the 90s, featuring Neil Young on a live version of America on a compilation CD to benefit Young's Bridge School charity, but the chances of them working together again are slim to say the least.