The Independence Exploit is a software based exploit that allows a PS2 to run unencrypted and homebrew applications (such as SMS) from the memory card. A description of what the exploit actually is can be found here.
The Independence Exploit is one of the techniques that can be used to trick your PS2 into running SMS. Normally your PS2 does not allow you to run unencrypted programs, and this protection is in place mainly to reduce the spread of copied games. When it comes to legitimate open source projects like SMS, the protection only stands in the way of running SMS. When installed, the Independence Exploit allows the user to run a specific application stored on the memory card. Often the chosen application will be an ELF launching program (a file browser) which then allows further applications to be launched. The Exploit itself is executed by inserting a chosen PS1 “trigger disc”, inserting a memory card with specially modified files on it, and then pressing reset. Tutorial of how to get the correct specially modified files onto your memory card can be found here.
The recommended approach is to use the tutorial to install an ELF launching program and an FTP server. Then you can use the FTP server to install as many other applications (ELF files) as you want. It also then allows you to make backups of all your saved games which is handy for those strange people who still use their PS2 to play games!
The exploit is significant to SMS users since it is one of the cheapest ways of running SMS, requiring no additional hardware other than a PS2, a memory card, and an original PS1 game.