The standard wiring pattern in USA (but not necessarily elsewhere in the world) for 110/120 AC is
black - hot
white - neutral
bare or green - ground
The hot wire brings the current to the outlet or fixture, often through a switch.
The neutral completes the circuit back to the panel.
The ground is a safety mechanism that, hopefully, will shunt the hot current to the ground (literally) rather than through a person if there is a short.
The terminals of a single pole switch (the only switch in a circuit and turning on/off one device or one set of devices) get the black wires, one on each terminal. Many switches also have a grounding terminal. This is not one of the switch terminals, but is usually a green screw on the metal strap of the switch frame, often on the opposite side from the terminal screws.
In general the white wires in a box should always be connected to each other, and never to or through a switch.
Similarly, the ground (bare) wires should be connected to each other and to the ground screw if the switch has one.
In your case, one set of wires (black, white, bare) are the source of the power, and one set (also black, white, bare) go to the fixture.
More complicated systems are possible with more colors, more switching choices and other configurations, but they don't apply in this case.
To work on this, make sure that the power is off (a non contact tester is best), before handling any wires.
Green and bare wires are always ground. Ground is a means of short circuit protection, and all metal parts in electrical systems and fixtures are grounded. If a hot wire shorts out and touches the enclosure for some reason, it flows to ground and the circuit breaker/fuse trips (typically this happens in nanoseconds).
Without ground, if there is a short, the metal parts become energized. Once you touch it, you form the path to ground and get electrocuted.
Old fixtures didn't always have this, but we got smarter after several electrocution deaths. Modern fixtures without exposed metal (eg made of plastic) or where any metal is mechanically insulated from the wiring may not have ground as well. Since this is meant to be hardwired, they won't have spent the time to make sure everything is isolated correctly, so absolutely it needs to be grounded.
So no, you should not just cap the ground off, you should get a new wire that includes a ground. If you are leaving the wire exposed (not in any walls), you would typically use SJO wire (aka "extension cord wire" -- which is flexible, and has all conductors in a round outer jacket) in this situation. Readily available at any hardware stores.
You'll also need a surface-mount junction box of some sort, with the SJO wire terminating into it through a strain-relief connector of some sort.
Best Answer
This might be due to loss of the neutral connection before the entrance panel. If the lights on some circuits get brighter and others dim when the heat turns on (or other high-current device starts), this would confirm that the neutral line is floating, i.e. the voltage on the wire, rather than being at nominal ground, is determined by the current drawn across each of the two main circuits.
If you have a multimeter, you can confirm that the neutral line is within a few volts of ground... not enough to give you a shock, unless you you contact wet skin or a cut. If the voltage goes above that for more than a second, you should contact an electrician to check and repair the neutral line.