Active versus passive: what does it mean, and why choose one over the other?
First, all the word "active" means is that some component inside requires power, either a battery or an external voltage source. "Passive" just means that no components inside need to be powered. Examples of active components include transistors, LED lights, and opamps or other IC chips. Common passive components include resistors, capacitors, and most potentiometers ("pots") like a volume knob.
Some instruments have an active/passive switch, and a few of them revert to passive operation if the battery dies. You can call them whatever you like, but they're active when powered.
There are three primary reasons to power a circuit: to boost the signal going through it, to lower the ouput impedance of the signal, and to isolate the signal from interference from other components. Lowering the output impedance increases the efficiency of the signal traveling along the cable. Poor efficiency can result in muddy tone, loss of highs, and low signal levels. Isolation can prevent "loading" (a drain of energy) caused by the instrument cable and other components. A classic example of this is with a two-pickup passive Jazz bass: when you set both pickups to equal levels, the overall volume of the bass drops a bit, and this is caused by the pickups loading each other down. Note that many "active" basses use a passive blend pot, so they can have the same problem of a volume drop at the center of the blend knob.
The active circuit that lowers the output impedance and provides isolation is called a "buffer". If the signal is boosted, then a buffer was almost always used for that job too, though buffers are not always designed for boost. All active preamps and EQ's include one or more buffers. Generally speaking you can only get boosted levels from an active device. But there is at least one system, by Villex, that claims to be able to boost the signal passively. The only way it is able to do this is by tuning a resonant filter to rob energy from the frequency ranges that surround the range being boosted. It works, it is an audible boost, but it has the limitation that the boost energy had to come from somewhere, and where it came from is a selected part of your tonal range getting reduced. That's not to say anything bad about that system though, it may sound great! I only mention it to clarify this one exception of passive boost.
So if active buffered circuits are so great, then why choose passive? Three main reasons: some people prefer the tone, some people prefer to do things "the old-fashioned way", and some people don't want to think about changing batteries. The tone question really is a valid concern. Many pickups sound noticeably different if you add a preamp. Also some preamps can introduce a tiny amount of compression or distortion--not everyone can hear it, but those that can, may not like it. Another big tone issue is that sometimes a person doesn't want isolation of the pickups, because the interaction between the pickups, the cable, and the amp can result in a pleasing tone with fat low mids and no harshness on top. The loading can actually shape the tone that way. However pretty much the same results can be achieved with an active instrument, just by adjusting the EQ on your amp.
What's the difference between "active pickups" and "active EQ" or "active electronics"? Most active instruments have passive pickups and an active preamp in the control cavity. When people talk about active EQ or electronics onboard, they are talking about that preamp. Active pickups are ones where the preamp is built into the pickup housing itself. The pickups are still passive at their core, but they have been wound and specially designed for optimal use with a preamp right there in the pickup body. The preamp inside an active pickup may be designed to do any of the functions of an external preamp: buffering, boost, or even EQ control. Active pickups may be combined with another preamp in the control cavity, but they don't need one. Again though, the majority of the time, just because an instrument is "active" does not mean it has active pickups.
So what's the deal with separate amp inputs for active and passive, or an active/passive input switch? Typically the active input or switch setting just drops the level of your signal going into the amp, to avoid clipping. This is because there was a time in the '80s and '90s when people generally thought an active bass always had higher output levels than a passive bass. It's not always true though! There are some very high output passive instruments, and some surprisingly low-output active ones, and everything in between. So what you need to do is use the "passive" input/setting for all low and medium-output instruments, regardless of whether they are active or passive; and the "active" input only for instruments that have such high output that they cause unwanted distortion in the amp.
What about DI boxes? Here are the pros and cons:
Pro passive: Doesn't require power supply. Doesn't clip harshly when hit with big signal spikes (instead, the saturation of the transformer sounds warm and tubey).
Con passive: Cuts signal level by -12 to -20 dB, so that if you're running to a mixing desk with inadequate preamps, the bass signal can end up being weak and/or noisy. Loads down passive pickups, resulting in a reduction/loss of high frequencies.
Pro active: Buffer prevents loss of highs. Doesn't cut signal level. May incorporate preamp-type features like EQ or a tube gain stage.
Con active: Requires power supply, either batteries or "phantom power" from the mixing desk. Cheaply-built ones may be noisy and unreliable. In some uncommon cases they may clip unpleasantly when hit by big signal spikes (depends on the specific design of the DI).
Essentially, if you have an active instrument (or other buffer/booster in the chain) then you can use either type of DI; you may prefer a passive one for its clipping characteristics, or just to avoid dealing with a power supply. If you have a passive instrument, you may be better off choosing an active DI in order to avoid loss of tone or signal level; however there can be cases where that isn't a problem, either because it sounds good to you, or it is feeding into a mixer with strong clean preamps. And whether you choose active or passive, it really is worth paying a bit more for one that has a reputation for being rugged, reliable, and noiseless.