The switch only needs to interrupt the power the positive (black) wire. The neutral (white) can be unbroken.
The ground wire though (should be uninsulated copper), should be connected to the ground screw (green screw) on the switch to protect against shorts in the system.
If you google how to wire a light switch you'll find good photos showing how.
That being said, it sounds like your box was wired by pulling a loop of romex into the box, instead of running a line in, then a line out (hence the unbroken neutral). Normally you'd run the romex (set of black, white, and ground wire in a white sheath) from wherever in the circuit it's coming from into the box, then cut it off with about 4 inches of extra in the box for wiring. Then you'd run more romex from the light to the box, leaving the same 4" tail in the box. After that you'd connect each of the black lines to a separate screw on the switch, the white lines together, and the copper lines together and then to the ground screw on the switch.