Current booking pertains to RESERVED tickets only. It has dedicated 'Current booking counters', separate from the regular general ticket counters. For General / Unreserved tickets, you have to go to regular ticket windows.
Railways' intent of providing Current booking is to sell those berths which remained vacant even after clearing all waitlists after chart preparation. So in true sense, current booking is like regular reservation only for all practical purpose. Such tickets shall be valid for that particular train only, and shall have specific Seat / Berth numbers reserved in your name (just like normal reservation)
Now...
more... coming to UNRESERVED / General tickets: They are issued at regular counters, they dont have name of train or seat no.s written on them, just names of Station A to Station B. So you can use them on any train running from A to B.
Since they r not linked to any particular train, they can be purchased anytime on travel date, irrespective of departure time of your train.