A self-extracting executable (or self-extracting exe) is a file that is usually created using a compression program. They are very much like a typical zip, ace, gz, tar, or hqx file. However, whereas these compressed file formats require a second application to decompress the files, a self-extracting executable can decompress itself without a second program.
The characteristics of this independent "mini-application" have their ups and their downs. On the plus side, is simply the fact that it is independent. You can give the file to anyone, and they can run it and extract the compressed files without requiring special software to do so. On the same issue, since the executable file is independent, it is required that it store the necessary decompression data and algorithms inside itself. This almost always means that a self-extracting executable will be larger than a simple compressed file such as a zip archive.
There are many applications that allow the creation of self-extracting executables, the most popular of these being WinZip (Windows), gzip (*NIX), and StuffIt (Macintosh). The programs take several files as input, and generate a single output file that is able to be run/executed on the operating system that it was created on.
Extracting the files in a Graphical User Interface (GUI) such as Windows, X-Windows, or MacOS involves simply double-clicking on the file. In a Command Line Interface, you can usually just type the name of the file to execute it. Depending on the application that created the executable, the files may immediately be extracted to the current working directory or location of the files. A more user friendly feature is to allow the person running the file to select a directory/folder to place the extracted files.
Self-extracting executables are useful as simple installation programs. A software developer can create a single file that is easy to distribute, and easy for the end user to use. They tend to be commonly used on the Internet because they are compressed, and therefore quickly uploaded or downloaded.