What is the meaning of "gap" in G1 phase?
The $G_1$ gap is the period between the last cell division and that start of DNA replication (synthesis). That's what "gap" means here. A period of time between two events. It's often described as time spent synthesizing proteins, but I prefer to think about it as the period before the signal to start synthesis lands. A cell can do any number of things in the meantime: synthesize proteins, grow, interact with other cells, move, differentiate, enter and exit $G_0$... 



Albert's has a good chapter on the overview of the cell cycle and on the control system