Apart from seeming a few, I don't know much about how gadgets work. Are they special applications, or regular apps with special styled windows?
I can't think of any app I use which has asked me if I want to install a gadget... I'd have thought any application which is typically left running all the time would be suitable... Skype/MSN for example. Am I misunderstanding the point of gadgets, or are they not popular?
Windows Desktop Gadgets are pretty much embedded Internet Explorer windows (if you use Visual Studio you can use Spy++ to find this out).
Like any widget system, they're very useful for displaying information, providing shortcuts or quick ways of performing repetitive tasks. There's a whole host of examples -- including some of mine, which I won't post as spam here -- at Windows Live Gallery.
The structure of a gadget is simply a folder, aptly ending with .gadget, containing a manifest document in XML format and at least one html file (referred to by the XML document).
An API is exposed to scripts running in the gadget in the form of a namespace, which provides access to methods not normally available to scripting environments.
To answer your other question, most applications with a permanent presence on the desktop already had the means to do so before gadgets came along on Windows. Most gadgets are created by hobbyist developers, but I've seen some instances of gadgets that came with applications (one of the CD/DVD creation software titles did it, not sure which). One of the setbacks is a lack of gadget to application based interaction/communication, which is difficult to do even with JScript's COM support. Windows 7 introduced an interface for adding a gadget to the user's desktop, which helps in at least one aspect.
After revisiting this answer I decided to look for the applications I'd seen installing gadgets. So far, I've found two, not linked because I don't want it to be regarded as spam:
Other reading: