Air traffic control radar is passive radar. For an airplane to appear on radar, it's transponder (xpndr) must be turned on. Each plane, while in contact with ATC, must have their transponder set to a four-digit code assigned by the tower: 0000-9999. In an emergency (if I remember ground school correctly), the pilot is supposed to "squawk" (or dial-in) xpndr code 7777. This will appear as an aircraft in distress on the radar controller's screen.
But what happens when the xpnder is turned off? The plane vanishes from ATC radar. ATC radar, as a passive system, requires that a return signal be transmitted from each aircraft being tracked.
If an aircraft MUST be tracked, and it's transponder has failed or been disabled, active radar must be used. Active radar "paints" it's sweep arc with radio waves and "receives" radio reflections and doppler frequency shift instead of relying on a return signal from the target aircraft. Weapon target acquisition radar and weather radar are also types of active radar.
As you might guess, Active radar has much shorter range and requires a lot more power to operate than passive radar. Plus, ATC does not have active radar systems at their disposal. To the best of my knowledge, only the military has active radar emplacements.
Hope this helped, and didn't cause too much confusion...