A hardline is a phone that has been modified by a redpill. I think volume 1 of The Matrix Comics has a story that shows a couple operatives rewiring a phone line to act as a hardline, or something along those lines.
From what I can tell, a hardline emits certain tones that interact with the code of a redpill's RSI, that allow for exiting the matrix or transporting around it. Since a bluepill is still hardwired to the simulation, the tones coming from a hardline would have no effect. If a bluepill picks up the receiver of a hardline, they probably won't hear anything and will assume the line is dead.
I would imagine each hardline would need to somehow be connected to a kind of broadcasting beacon in the real. This explains why hovercrafts need to be at "broadcast depth" in order to hack into the matrix. The ship gets within range of one of these beacons, and then broadcasts a signal. So in order for a new hardline to be created, it needs a beacon in the real to connect to.