I'll try to make the following as simple as possible but it can be hard to explain without using the terminology for weavers. But here we go:
Satin is just simply a weave structure-- meaning, it's just a specific way to weave. There are three very basic weaving structures that (most) all other weaving is based upon. Those three are: plain weave, twill weave, and satin weave. (I'll get to satin in a minute)
Plain weave is really basic. Every culture on earth that has weaving has plain weave in there somewhere.  This picture shows basic plain weave.
The fibers running up and down are called the warp. The fibers running from left to right are the weft (think weft rhymes with left).
As you can see in the picture, the weft (left to right fiber) goes over one fiber then under one fiber over and over again. And each row alternates so no fiber is always jumped over or slid under. You don't want to leave anyone out. See that? That's what locks those loose fibers together into one sheet of fabric. Without them locking together like that, all you'd have is  this and that's not fabric.
The second weave style, twill, is very similar to plain weave. The only difference is that instead of the fibers going over one, under one, it kind of jumps. So, it can look like  this or  this . These are both very basic twills. In the first one, the white weft jumps over two fibers then slides under two fibers. In the second one, the darker weft jumps over two fibers then goes under one fiber. But they're both twills.
Twill fabrics are generally stretchier than plain weave 'cause the fibers can move around a bit without pulling out or falling apart. They're flexible where plain weave is very sturdy and doesn't like to move at all. Twill fabrics have to be careful, though 'cause if the weft tries to jump too far, the fabric can become weak. Those weft fibers can snag easily if they're too long. And that can make the fabric fall apart, which isn't good at all.
Now, the third main type of weaving is the satin weave, which is like a brother to a twill but is more daring. Satin jumps over four or more fibers, goes under one, then does it again. It looks like  this . See how the blue weft jumps over many threads there? That's what makes satin a satin.
Satin is shiny because there are so many fibers floating and the light is reflected more evenly from them. It also helps that modern satin is made from shiny fibers like rayon and silk. Often, the weft fiber will be a shinier fiber than the warp so the shininess is more obvious, but it's the way the fibers are woven that make it a satin; not the fiber types.
You could make a satin-weave scarf out of wool. It would be shinier than a twill-weave scarf but it wouldn't be as shiny as a silk satin.
Which brings us to the difference between satin and sateen. Satin, as I said above, is a weave structure like twill and plain weave. Sateen, though, is just a satin fabric made from cotton (or some other short staple fiber). Generally, cotton sateen is made when the weft goes over four fibers and under one fiber as anything longer than that and the cotton makes it weak. Cotton is not as strong a fiber as wool or silk.
Satin is wonderfully shiny and slippery but it is not as durable as plain weave or twill. Even when made of silk (the strongest fiber), the jumping weft means it lacks stability.
If you have any further questions about weaving-related things or if any of the above terms weren't clear enough, please let me know.
Source: I'm a weave structures and print pattern textile designer and a weaver myself. I focus on twill-variants, damask, brocade, and shadow weave. I also have a very strong love for pictorial tapestry weaving.
edit: For a little more non EILI5 info: there are literally hundreds of variations of pretty much everything you can weave, even plain weave. I picked two twill examples but there are many others and twill alone has many sub-categories of weaving styles. Like random walk, corkscrew, shadow weave, and a bunch of others. Satin, too, can jump more than four warp, so there's a lot of variation there as well. These three types of weaves can be combined to make damask, brocade, and overshot, to name a few. The above is just the most basic I can make it.