Active gives you a preamp on the bass itself, almost always with an equalizer to boost or cut bass, treble, and often midrange levels. Basically, it gives you more tone control on the bass itself than passive basses do, which for tone shaping usually just have a tone knob (which is basically a low-pass filter and not an EQ).
Using my bass as an example, it’s got five knobs and a switch. The switch is an active/passive control. One knob is volume and another is pickup blend. The other three knobs are bass, midrange, and treble boost/cut while it is active. When it is passive, the treble control becomes a tone knob and the bass/midrange knobs do nothing.
The advantage of active over passive is the increased tone control on the bass. The disadvantages are the battery, and also potentially noise from the preamp (though with good basses this is usually not an issue.)