Pickups consist of a magnet with copper wire coiled around it. When the vibrations of a bass string disrupt the magnettic field of the magnet, small voltage fluctuations in the copper coil are produced. These fluctuations are then transmitted to the bass amp, then amplified and translated into sound. 

Active Pickups:

"Active pickups", are battery powered pickups. In an active pickup you have the same basic construction as passive pickups. ,but the makers add a built in preamp powered by an on board 9 volt battery. Now your output level is being boosted by the preamp instead of the pickup construction. This allowed the designers to lower the magnetic pull on the strings to allow longer sustain. Also this allowed the makers to build custom tailored preamps to EQ the pickups in a more controlled and even way. Some active pickups have a wider more level frequency response than passive pickups. And this is perceived as being not as warm sounding.

==> Pro: preamp's are built in.
==> Con: You spend about $5.00 per year for the battery, which isn't worth mentioning as a con

Passive Pickups:

"Passive Pickups", for the most part are not battery-powered. They are made hotter by increasing the magnetic energy from the magnet and also increasing the copper wire wrap. What this does in effect is boost the signal from the strings vibration. Passives pickups can be boosted / altered through a separate pre-amp's EQ attached to the bass amp, or an EQ pedal.  Passive pickups have been modified through construction. Manufacturers change frequency response, output level, etc... by using different types of magnets, wire, even different plastic spools the wire is wrapped onto the magnet.

==> Pro: You can add a preamp to your bass's wiring system. EMG has a preamp available.
==> Con: You experience "string drag", essentially a loss of sustain as the strings vibration is dampened by the magnetic field.