What does "gig" stand for in "It's your gig"?

It's an extension from it's your call/show/shout/turn, where gig alludes to the now-popular sense of a live performance by or engagement for a musician or group playing popular or jazz music.

Probably in many contexts your gig could be replaced by your turn [to perform now], but it's also used in the call sense (i.e. - as you're the "star of the show", you have free choice over the details).

Gig has often meant gigs for musicians, as Shyam said. However, it has today evolved to slang for almost any kind of job or reservation on your time. From this context, "it's your gig" could be a congratulatory remark saying that you got the job, or it may simply signify or reiterate your ownership of the said job.