If you have a full size propane model, you'll want to first turn all of the burner control knobs (including the side burner, if present) to the off position. Then, turn off the valve on the top of the propane tank.
If you have a portable propane model, such as a Q series or Go Anywhere series model, and you're using the grill with a disposable propane canister, simply turn the control knob to the off position. If you're using the grill with a standard 20 lb tank via Weber adapter hose 6501, turn off the control knob and then turn off the valve on the top of the propane tank. If you intend to transport the grill or store it indoors when it is not being used make sure to disconnect the disposable canister if that's what you're using with the grill.
If you have a natural gas model, you'll want to first turn all of the burner control knobs (including the side burner, if present) to the off position. Then, turn off the shut-off valve on your natural gas supply line.
Always turn off the fuel source to the grill when it is not being used. Leaving a fuel source open/turn on is a safety hazard. For more info, check out this article.
Propane tanks should never be stored indoors.