A cast recording is a recording of a stage musical that is intended to document the songs as they were performed in the show and experienced by the audience. An original cast recording or OCR, as the name implies, features the voices of the show's original cast. A cast recording featuring the first cast to perform a musical in a particular venue is known, for example, as an "original Broadway cast recording" or an "original London cast recording".
Cast recordings are (usually) studio recordings rather than live recordings. The recorded song lyrics and orchestrations are nonetheless identical (or very similar to) those of the songs as performed in the theatre. Like any studio performance, the recording is of course an idealized rendering, more glossily perfect than any live performance could be, and without audible audience reaction. Nevertheless, the listener who has attended the live show expects it to be an accurate souvenir of the experience.
The British were the first to make cast recordings, and they were also the first to make original London cast recordings of shows that had already opened on Broadway, but had not been recorded with their original Broadway cast. This led to the odd situation of having, for example, a 1928 recording of the London cast of Show Boat, but no recording with the actual 1927 Broadway cast, and a recording of the London cast of Sigmund Romberg's The Desert Song, but not of the 1926 Broadway cast - even though both of these shows are Broadway musicals, rather than British ones.
Prior to the development of original cast recordings, there had of course been recordings of songs from musicals, and collections of several such songs, and recordings of songs performed by cast members; but they were recordings of songs, that is, they were recorded as stand-alone show tunes rather than representations of a (more or less) complete musical.
The first American original cast recording as we know it was an early experimental LP of program transcriptions of selections from The Band Wagon, a 1931 revue starring Fred and Adele Astaire. It was not widely released.