How do I run a web application "offline" on my Android? I have a web application written using HTML5 that leverages the application cache for "offline" access.

On my iPod touch, I can go to the web while online, click the "+" icon, and set a shortcut on my home screen. This will launch the web application full screen and run the app, even when there is no Internet connection.

Is there a similar process to use on my Android device, or will it vary by maker?
Offline data saving is supported on Android, you can see this if you go into your browser, select Settings, Advanced Settings, Website Settings and you can see which websites are using your offline cache, as well as which are using other advanced access like GPS.

To create an icon for a website on your home screen: once you've created a bookmark in your browser for the page, you can go to your home screen, long-press on a bit of spare space, select Shortcuts, Bookmark and pick that one. That should give you an icon straight to that page.