I used it in a computational-media class a few years ago, where it was the main language used for the examples and projects. Since I was a CS student and already comfortable in a bunch of languages (much of the class wasn't), I wasn't really the target audience, but I thought it was reasonably well done for what it is. You can get some graphical stuff happening on screen in only a few lines of code, and the IDE automates building a Java applet that you can ship just by copying a directory to a webserver, both of which, I think, increase the satisfaction to someone just starting out with programming, because you can immediately see the effect of relatively simple programs. It's actually a lot nicer than Java for simple 2d graphics stuff, because it gets rid of a lot of the boilerplate, and includes some decent built-in primitives. And you can escape to Java if things get complex.

I've used it a few times since then as a rapid-prototyping tool for 2d graphics stuff, mostly interactive graphics toy type of stuff, and I think it's pretty good for that. HTML5 Canvas is getting to the point where it'll probably replace it for that purpose for me, but when Processing came out in the early 2000s, there wasn't much comparable when it came to getting some interactive graphics working in a web browser in <10 lines of code.

I haven't used Processing explicitly but from what I can tell the Arduino's Wiring language is pretty much the same thing (it uses the Processing IDE and from what I've read it seems to be a superset of Processing). I use Wiring when playing around with my Arduino but generally rewrite things in assembly if I take a project further than just messing around.

I have to say, the name put me off. It'd be like calling a car the Ford Driving. Or Starbucks launching a coffee beverage called Drinking. It reminds me of http://paulgraham.com/javacover.html