From Rolling Stone:
“Desperado” was the title track of the Eagles’ second LP, a concept album about outlaws in the Old West. “In retrospect, I admit the whole cowboy-outlaw-rocker myth was a bit bogus,” Henley said in 1987. “I don’t think we really believed it; we were just trying to make an analogy.”
Don Henley stated in the notes written in the booklet to Eagles’ 2003 “The Very Best Of” compilation, that Desperado was a piece of a song that he had written in 1968. He said, “It was called something else, but it was the same melody, same chords. I think it had something to do with Astrology [Chuckles].”
From allmusic, reviewing the album:
On Eagles, the group had already begun to marry itself to a Southwest sound and lyrical references… All of this became more overt on Desperado, and it may be that Henley, who hailed from Northeast Texas, had the greatest affinity for the subject matter. […] What would become recognizable as Henley’s lyrical touch was apparent on those songs, which bore a serious, world-weary tone.
Don Henley’s rough voice is one of experience, and it helps make “Desperado”…memorable. The title track features Frey’s slow, moody piano intro and Jim Ed Norman‘s cushioning strings that don’t dare seek prominence.