Backline is another word for gear or equipment. When a gig is booked, sometime before the concert, the agent, manager, or band should liaise with the promoter, venue, or club to discuss the backline needs of the band - the promoter will need to know what the band is bringing with them in terms of instruments, mics, amps and so on, and what they will need the promoter to provide. The promoter will also need to know the technical specifications (tech spec) for the band.
Exactly whose job is it to provide backline really depends on how the show was booked and what size the show is. If a band has booked a show directly with a venue that they are promoting themselves, then the band needs to figure out what the venue can provide for them and make arrangements for everything else themselves. If a band books a show with a promoter, the promoter generally will make sure the backline is in place (though if the promoter has to rent equipment for the show, they will usually charge these costs back to the band). If an agent has booked the show with a promoter, then the agent and promoter should handle the backline negotiations.