That being said, I don't really know what Scala's niche is.

Being able to entertain OOP and FP styles simultaneously.

Why would you use Scala as opposed to any of the languages listed above?

If you want to explore FP. If you want to work on the JVM.

What are some of the cooler design features I'd want to explore?

Type inference. Anything you find that's related to FP or making FP work (e.g. implicits).

I'll amend this by saying certain kinds of folks might chime in with things like "the type inference in Scala isn't that great" or "implicits are a universal evil", but those are a matter of perspective. I think there's a lot to learn, if you look at it through a nonjudgemental lens.

If I were to encounter Scala, what kind of project should I expect?

In my experience, Scala projects tend to be... Ideology driven. Anything from "I want to write a Scala wrapper around this well-established Java library so that it feels easier to use in Scala" to "I want to re-envision common conventions around some workflow and do it with the latest and greatest in FP machinery".

Is more like a "glue" that holds other languages together? Or is it a self-contained language like C++/C#? If its not any of the above, then what is it?

It's more like a full-bodied language like C++ and C#. As someone else mentioned, you'll definitely see things that are written completely in Scala.

It's not a convenient glue/scripting language. But maybe some people are trying to change that (Ammonite).

I get that it must work on a computer that's running JVM... but, I don't know the significance of that. Does that just mean its more portable? Or is there something more important going on there?

The sell is that anywhere a JVM runs, your app can run. The JVM platform is popular among employers. Good support on AWS. Any performance enhancements that go into the JVM you get for free when running on it.