Cycles = Night and Day periods.
The game moves in cycles of NIGHT and DAY. While there may be exceptions, night usually happens first, then it alternates, worded like this:
Night One, Day One, Night Two, Day Two, Night Three, Day Three, etc... or N1, D1, N2, D2, N3, D3, etc ...
During night, conversation on the thread is usually limited, although BTSC conversation (usually between baddies) is flaring like a live wire as people decide the night actions they're going to do. Most Mafia games have many night-active roles and quite a lot of things can and do happen in a single night. Not all of it goes into the night post, and much is kept hidden and behind the scenes (for example, hosts may not show in posts who spy-roles spy on).
Usually hosts set time limits on night (18-24 hours is a typical range), though if they get all the night PMs in before that, marked as final (usually sent in red e.g. N2 action: Spy X), they will post the night post earlier, or as soon as they can, sometimes even a bit after the deadline. This is a good standard, but each host chooses and publicly sets time limits in the thread.
Days for a normal-length mafia game are always 24 hours, ALWAYS. Regardless of whether the host is there or not, the day ends. Not earlier (except for No contests), and not later. When the host comes around, the results at the end of the deadline are taken into account, anything past the deadline is discarded.
At the end of each night & day, the host makes a "night post" or "day post", which chronicles (usually in story form) the events that happened that are now publicly known during the night/day. Hosts then update the roster with new deaths & developments and announce the following day/night phase and when it ends.