An AppImage is a type of cross-distribution packaging (or bundling) format. It is essentially a self-mounting (using Filesystem in Userspace, or FUSE for short) disk image containing an internal file system for running the application it provides. This internal file system includes all the binaries and libraries required by the application that cannot be reasonably expected to be provided by the base operating system (that is, the Linux system on which the AppImage is being run).
Usage
To be run they first need to be marked executable (with chmod +x <AppImage>, where <AppImage> is the file name of the AppImage, including its file extension) and then run
with ./<AppImage>. Either that or clicked/double-clicked in one's file manager.
AppImages available for download
A full list of AppImages available for download can be found at the probonopd/AppImageKit GitHub Wiki and The Hornery.