In most eeg readings passive electrodes are used. They are cheap, easy to manufacture and maintenance. They require however special skin preparation in the spot under electrode, and special pastes are also required to lower impedance between electrodes and skin. Those requirements are usually acceptable, but not always welcome.
Active electrode is an electrode that requires no skin preparation. It is possible with preamplifier placed very close to skin (inside the electrode). High dry skin impedance can be omitted by using amplifier with very high input impedance.
Another reason to use active electrodes is safety. Because the dry skin-electrode impedance is very high (a few megaohms) then the isolation barrier is much higher and a chance for an electrocution is much smaller comparing to passive electrodes.