Before we get into the nitty-gritty of Adobe AIR, it helps to understand a few key terms and break down what it means to be a "cross-operating system runtime" that can run desktop applications.
AIR is, first of all, a runtime engine at its most basic. A runtime engine is simply computer software that other applications need to use in order to run properly; it translates language within a program into machine language, the simple, lowest level language (essentially 0s and 1s) the central processing unit (CPU) can understand. Programs that run on Java, for example, require the Java Virtual Machine runtime engine installed on the computer. Without it, your computer wouldn't be able to make use of the same graphical user interface (GUI) that you're used to using now. A GUI, also known as a human-computer interface, is the combination of windows, icons, text and menus that we can change with a mouse or a keyboard. Even your operating system can be considered the mother of all runtime engines, since every application on your computer depends on the processes of the operating system.
So when Adobe describes AIR as a "cross-operating system runtime," they just mean that AIR is a runtime that can work on any computer, regardless of the operating system. Different operating systems use different languages; a program running on Mac OSX, for instance, won't look the same as the same program running on Windows XP. Applications built with Adobe programs such as Flash or Flex will use the AIR runtime to display -- because AIR comes the with the open-source WebKit HTML renderer, Web-like content can display on the desktop. That's the big difference between something like a Flash player and AIR: While Flash works over the Internet and in your Web browser, applications running off of AIR are based on the desktop and don't necessarily require a browser to work.
Finally, what are the RIAs that Adobe AIR will help build? An RIA is a broad term coined by Adobe that describes a Web application that provides an engaging, connected experience for users. In general, they're meant to provide the best aspects of both desktop applications and Web-based applications and are developed with programs like Adobe Flash or Flex -- Web sites like Flickr or Google Maps are examples -- sites that offer lots of interaction but are located on the Internet. RIAs are easy to use but difficult to program, so many hope the release Adobe AIR will make such development processes smoother.
That's what AIR does for you computer applications, but what does it do for developers, businesses and everyone else?