Note up front: I am not an expert on this subject, as I am not much of a fan of pickups on ukes in general. I prefer the more natural sound of a good microphone placed near the instrument instead. But I do understand that there is a time and a place for pickups and they can be quite convenient in some situations. But I am also a physicist who is fascinated by sound (and the reproduction thereof) so I did some digging in the past, so I will pass along my two cents. But since I am not actually a sound engineer nor a professional performer, that is probably what this is all worth, about two cents...
Broadly speaking, you have two classes of pickups: active and passive. They both work using similar technology, but with some extra wrinkles. Let's start with passive and build from there.
A passive pickup doesn't have a battery onboard and usually (maybe never? I don't know of any off the top of my head) have any extra knobs, wheels, or controls. It has a disk (or multiple disks) that attach inside the instrument, underneath the saddle. These disks are piezoelectric. They contain a crystal that puts out a very low voltage change when they are compressed by vibrations. This very small voltage change is what goes out to the jack and ultimately to your amplifier. And that last word is kind of key: a passive pickup assumes you are going to run the signal to an amplifier in order to amplify the signal enough to get a good volume. This could be as simple as a decent guitar amp. To get the best sound possible, some folks will run the passive into a pre-amp first, where they will adjust the tone (think of this as a crude equalizer that ramps up the lows, mids, and highs to your liking) and the volume level before running that signal onward to speaker system. (This is also what the knobs on most guitar amps are attempting to do...)
The advantage of the passive type pickups are that they are dead simple and the signal they deliver is usually pretty clean. Not a lot of electronic noise (before you run it into your amplification "chain".) Basically, every penny of what you are paying for a passive is going into the quality of the pickup. The downside is that you might want a pre-amp (which can be pretty expensive if you are looking to get a good one. But you only need one to use with all of the passive pickups on all of your instruments...)
An active pickup is basically a passive pick up with the pre-amp built in and powered by a battery. Some have a knobs to control tone and volume, some don't. In general (and I really mean that, take this with a grain of salt) active pickups have more electronic noise in their signal than a passive one does. When you buy an active pickup, the cost is not just going to the pickup, it is going to the pre-amp too. So if you buy a $150 active, not all of the $150 went to making a really good pickup, some went to the rest of the electronics. This is why, in general, similarly priced passive and active pickups won't really be of similar quality.
The regular MiSi that a lot of folks use is an active pickup, but with a rechargeable power system (I think it is just a capacitor, not a battery, since it charges so fast.) Instead of being a disk of piezoelectric material inside the instrument, it is a strip that is inserted under the saddle itself and then run into the electronics inside through a small hole drilled in the bridge. They work pretty well, but you have sand down the saddle to make room for that strip and you have to drill a hole.
Then there are the weird ones, like the Anuenue Air-Air pickup. It couples a piezo pickup with a tiny microphone. Both are mounted inside the body, and you have controls to mix the signals from the two sources to get a really nice, natural sound. Word is that they sound fantastic, but they are pretty pricey. I believe that the MiSi Air does works the same way.
I mostly don't like pickups (particularly active ones), because I think they make most instruments sound more similar. In other words, the sound of a high end koa with an inexpensive active pickup isn't *that* much different from an inexpensive laminate with the exact same pickup. They just sound "same-y" to my ears. If you go for a good pickup, this becomes less true, and if you go for a good passive pickup it is better still.
The most "natural" sounding pickup I have personally run across is the K&K Twinspot. It is passive, and has two piezo pads; one goes inbetween the G and C, and one between E and A underneath the saddle. This is the pickup that Mya Moe used (is still using?) and is installed by Aaron Keim at Beansprout. Run it into a decent pre-amp and it sounds more like you have used a mic than a pickup. It still sounds like you can "hear" the wood, if that makes sense.
K&K also makes the Aloha, which as I understand it is the Twinspot but with shorter wires (since the Twinspot can be put in a guitar.) I can't swear this is true, but it is what I have heard, so I am passing it along.
