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Grounding is important for safety and proper operation of the coil. The Tesla coil should have two separate grounds.

The first ground is the house or building ground. This is the green wire in the electrical outlets.

The second ground is called the RF ground. This ground should be connected to a metal grounding rod that you pound into the ground. Although there is already a ground rod installed outside homes and buildings, you should not connect to this rod because it's connected to the house or building ground. You'll have to pound in your own ground rod. The grounding rod should be as close as possible to the Tesla coil, and as far from the house or building ground rod as possible. Generally 6 or 8 foot minimum depth is recommended, but it really depends on soil conditions and other factors. Deeper is always better. Several shorter ground rods can be placed around the Tesla coil if a single rod can't be used. If the ground is very hard, rocky or dry, you can place a piece of metal plate, chicken wire or mesh under the Tesla coil and use it as your RF grounding. The radius of the plate or mesh should be approximately equal to the height of the secondary coil and top load. This type of ground is called a "counterpoise". If you don't have access to a ground rod or counterpoise, you can connect to a cold water pipe. As a last resort, if you're on a ground floor that's at least semi-conductive you can wet the floor and put a layer of aluminum foil down. This is not recommended for safety reasons and you'll have to use this method at your own risk. Braided copper wire can help the conductance of the RF ground, but regular wire will work just fine. Wetting the ground around the ground rod before running the coil helps conductivity to the earth. Be careful not to damage underground utilities when hammering in the ground rod. Poor RF grounding may not have any apparent effect on the Tesla coil - or it could cause reduced arc length, arcing up the secondary coil, or arcing between the primary and secondary coils.
warning

Proper grounding is extremely important.

Proper grounding of a Tesla coil has been debated for quite some time. The general consensus is to connect anything you will touch during operation of the Tesla coil to the house or building ground. Anything that may be struck by an arc, or that may experience high voltage transients, should be connected to RF ground. The general idea is to pass all high voltage generated by the Tesla coil to the RF ground. We also need to prevent any high voltage spikes making their way into the house or building wiring. The NST seems to be a good boundary between the house wiring and the Tesla coil wiring because the primary and secondary windings are basically isolated from each other. Therefore, anything connected between the house outlet and the NST primary (variac, control panel, line filter) should be grounded to the house ground. The bottom of the secondary coil, the primary strike rail, NST protection gap and filter should be connected to RF ground. Grounding the NST case seems to have caused the most confusion. I recommend connecting it to the RF ground because it's usually more likely to be struck by an arc or experience a voltage spike.
warning

Using separate grounds on the NST input and NST case can cause a potentially dangerous situation if the NST is connected to the house wiring, the RF ground is disconnected and a short develops between the windings and the case. In this condition someone could be shocked if they touch the NST case because the case in not grounded.

There is no absolute rule for proper Tesla coil grounding. It's your responsibility to understand the electrical principals of grounding, seek advice and information, take your situation into account and anticipate potential dangers.