What it means
A ciggy is just a cigarette, the cosy little shortening you use when you are not being formal about it. You nip out for a ciggy, bum a ciggy off a mate, or stand in the rain having a quick ciggy outside the pub. It is everyday British and Aussie talk, friendly and casual, the same smoke by a softer name.
Usage examples
"Just popping out for a ciggy, back in five, save us a seat by the window."
"He nipped outside for a quick ciggy between meetings."
"Can I bum a ciggy off you, I left mine at home."
"Got any ciggies left or did you rinse the whole pack on the walk over?"
"She was out the back with her coffee and a ciggy, hiding from the chaos inside."
Where it comes from
It’s a straightforward clipped form of cigarette with the affectionate -y ending that shows up all over British and Australian English. That pattern gives ordinary words a more relaxed, matey feel, so ciggy landed as an easygoing everyday way to mean a cigarette in casual speech.