"Active" is generally too vague to mean anything other than that there is a buffer/preamp stage that requires a DC power source. Often, it means there is an onboard preamp with equalization. To characterize the specifics, however, there are a number of uses for buffers and preamps on basses.
1. Lowered output impedance. The use of long runs of instrument cable creates a parasitic capacitance that interacts with the resistive component of the signal impedance to create a low-pass filter. Lowering output impedance thus shifts the frequency cutoff point up, reducing the "tone suck." This is one reason why people often incorrectly assume that active components have an inherent tonality.
2. Constant input impedance. The pickups will see a constant input impedance regardless of what you plug the bass into. This means that various amps, effects, cables, etc. will not influence the part of the circuit between the pickups and preamp input.
3. Multiple buffered inputs. This is relatively rare, as most preamps have a single input, but in some cases, such as when mixing pickups of significantly dissimilar output impedance (Piezo and magnetic pickups, for instance.), it may be necessary that each source be given its own input that is not influenced by other parts of the circuit. This may also be useful for things like buffered blends, and volume adjustment of individual input sources.
4. Equalization. This is what most people are instantly familiar with. Two, three, or four band bands of equalization with or without adjustable frequency centers is common.
5. Fixed voicing. Many players prefer a tonally transparent preamp, but some preamps may be designed to apply a particular voicing, even with the EQ pots set flat. This may be the signature of a particular manufacturer, or it may be a design choice based on the use of pickups that are naturally lacking in some regard, or in need of tonal compensation from a preamp. EBMM's Stingray pickup and preamp pairing comes to mind.
6. Gain boost. Some preamps provide a boost of gain. When a preamp offers this feature, it is often adjustable with a trim pot on the PCB module. Many people incorrectly assume that "active" is synonymous with "louder," however, and this is not necessarily true. Preamps may be unity-gain.