This will be more or less summarising what others have stated already, but broadly speaking there are two broad categories of "electric" ukuleles that are regularly seen.
1. Nylon string, piezo pickup
This includes both ACOUSTIC-ELECTRIC ukuleles and SOLID-BODY ukuleles.
They both use piezo pickups, which are the pickups of choice in most acoustic-electric instruments.
They produce a signal for the amplifier by converting the physical vibrations of the strings (usually via a piezo bar underneath the saddle, or a transducer on the soundboard) into electric currents.
These are designed to closely reproduce a clean acoustic tone of an instrument.
Acoustic-Electrics are basically just normal acoustic ukuleles with a pickup installed.
Solid-body electrics are dedicated to be played electrically and produce little acoustic sound but usually produce a superior sound through the amplifier.
The solid structure of the body improves the amount of vibration energy transferred to the piezo pickup, resulting in greater volume, sustain and clarity than a similar quality Acoustic-electric.
Furthermore, solid-body electrics will always be less susceptible to unwanted "feedback" produced at higher amplification settings compared to acoustic electrics.
Example of an acoustic-electric ukulele plugged in (played by me):
https://www.youtube.com/watch?v=LDEOs_hpqVg
Examples of a solid-body electric ukulele (nylon strings):
https://www.youtube.com/watch?v=tNIimZxy3iw
https://www.youtube.com/watch?v=3tqY6OPtJzQ
As you can see, solid body nylon string electrics can be made to sound like an electric guitar by using electric guitar amplifiers and effects.
However, they are not quite ideally suited for that.
You can do the same with an acoustic-electric, but it won't sound as sustained and the amplified sound may be susceptible to feedback.
In summary, an acoustic-electric is
designed for faithfully replicating natural acoustic sound.
A solid-body with a piezo pickup has the same purpose, but is a bit more flexible with using it as an "electric" instrument.
2. Steel strings, magnetic pickups
These are what I would call "true" electric ukuleles, when compared to the guitar family of naming instruments.
These are basically 4-stringed electric guitars that are the size of an ukulele and are tuned like an ukulele.
Basically this allows you to use the ukulele exactly as you would use an electric guitar.
They use the same strings as electric guitars, hence the tension and feel of the strings will feel like an electric guitar, rather than an acoustic ukulele.
In contrast, you could say nylon-string ukuleles feel like a classical guitar.
This kind of electric ukulele will sound the best when you're wanting to make electric guitar sounds.
Examples (by humble me):
https://www.youtube.com/watch?v=lkpn5bN2l1M
https://youtu.be/FwdW-IEqJ4M?t=29