A&R is the department of a record company that finds and develops bands, songwriters, or musicians. More often than not A&R reps help out with a bands artistic and commercial marketability. Think of the A&R Department as the middleman between the band and record label.
A&R Reps from major or larger record companies may only listen to solicited demo submissions from musicians or bands. A solicited submission is one that a music company only accepts from known or reputable contacts with whom they've networked in the past. Smaller record labels may accept both solicited and unsolicited (anyone can send demo) demo submissions. A great resource to tell what record companies accept unsolicited and solicited submissions are the Songwriter's Market books (See books below). They give you information on what record companies may need with a demo submission and how to submit your band demos.