That Wiki article really is lacking in information. I'll expand on it a little.
So there are pretty much two kinds of solid axles in cars and trucks. The semi-float and the full float. Both are straight axles with a differential somewhere in the middle, shafts inside the axle tubes that drive the wheels on either end.
The simpler design is the semi-float axle. We'll start with a picture of the important end.
In this picture, you see the end that the wheel attaches to. On the right hand side is the end of the axle tube with a 4 bolt flange welded on. There is a stamped steel plate with 4 studs on it. This is the bearing retainer plate. Just inside the bearing retainer plate is a wheel bearing. That wheel bearing sits inside of the pocket at the end of the axle shaft. The retaining plate holds the bearing in place once it's bolted to the flange on the axle shaft (along with the brakes, which are also bolted to the 4 bolt flange at the same time).
Sticking out to the left is the flange on the axle shaft. It has 6 wheel studs sticking out of it. Once the brakes are put in place, those studs are what you use to hold the wheel on.
In this setup, the axle shaft is responsible for both making the wheel spin and for holding the wheel in place.
In some configurations, the wheel bearing is pressed on to the axle shaft. The bearing retainer plate holds the bearing in place and the bearing holds the axle shaft in place. In other configurations, the axle shaft has a groove cut into it on the other end and a "C" clip is put around that groove inside the differential and that clip holds the axle in place.
This is called semi-float because the outside of the axle shaft carries all the weight. Because it is carrying all of the weight it is not "floating". On the other hand, the inside end of the shaft just "floats" inside the differential. There's no weight at that end.
The full float axle is different. We'll start with a picture of the business end of the axle tube itself:
This axle has a similar 4 bolt flange to the semi-float. But it's only for holding the brakes on. The outer end of the tube is a precisely machined spindle. It's hollow in the middle so that the axle shaft can stick all the way through it. A wheel hub is attached to the spindle with a few bearings that are held on by nuts that thread on to the end of the spindle.
I couldn't find a convenient picture of a hub specifically for a rear axle but it's similar to one for a front:
This picture has two hubs (one old, one new). They have brake discs attached to them but they don't necessarily have to have them. You can see the wheel studs sticking up and the races for the outer bearings. (also, these hubs have splines in the end which I'll cover later) These are slid onto the spindle and a nut with a locking device of some kind is threaded on to hold the hub in place.
After the hub is installed, the axle shaft is slid through the end of the hub and it attaches in one way or another to the hub once it's all the way in.
Here's a picture of a full float axle fully assembled with brakes and the axle shaft not fully installed:
In this case, the full float axle has a small flange on it and you bolt it to some holes in the end of the hub. That way, the hub and the axle shaft turn together as one unit. The hub can be splined like the ones a couple pictures above. In that case, the axle shaft will be splined as well and you place a puck with splines that engage with both the hub and the axle shaft to tie them together. But the shaft with the small flange and bolts is by far the more common setup.
All by itself, the hub holds the wheel in place. All of the weight is held by the hub. No weight is applied to the axle shaft. In fact, you can remove the axle shaft without even taking the tire off (assuming the wheel doesn't cover the outside of the hub). The axle shaft is only responsible for making the wheel turn. Hence it is "floating" on either end. That's where "full float" comes from.
The semi-float axle is much cheaper to manufacture so it's much more common in cars and light trucks. But the full float axle is much better at carrying very heavy weight. Because of that, it's the only setup you find once you start to reach the 3/4 ton and heavier trucks. In the big rigs, it's almost impossible to find a semi-float axle.