Like M0ses said, what you have are metal strings vibrating over a magnetic coil (the pickup). That generates a small electric current in the pickup which perfectly matches the frequency the string is vibrating at; the cable then carries that current down to the amp to be amplified and retranslated into sound via the speaker.
Most bass pickups are either single-coil or humbuckers; in general, single-coils have a little more "bite" to them but a single-coil by itself will also generate a low humming noise; that's why jazz basses have two of them, if they're both on full they cancel each other out and don't have a hum. A humbucking pickup doubles the coils so that there's no hum in the first place; they also have a somewhat rounder, warmer tone with less bite in it. Sometimes you'll see a bass with a coil tap, which means it has a humbucking pickup but you can isolate just one coil for the single-coil sound by pulling a knob or flipping a switch.
There are some guitars with active (= powered) pickups, but hardly any basses have them. 99% of the time "active" bass means that a preamp has been added to the instrument; basically a tiny amplifier that can increase the signal and, more importantly, control its high, middle and low frequencies before sending the signal on to the actual amp. It needs a 9v battery (sometimes two of them) to power the preamp.
As others have said, there are a gazillion engineering variables that give each brand of pickups their own individual character; thickness of the wire, how many windings, etc.