DSi Homebrew Setup
From GameBrew - a wiki dedicated to Video Game Homebrew.
To get your Nintendo DSi able to run homebrew you should follow the guide on dsi.cfw.guide. This guide is well maintained and easy to follow.
Do not follow video guides from sites such as YouTube. They are harder to update and thus often out of date.
Once you're done you can check out the List of DS homebrew applications page to find homebrew to run!
Note: Many older homebrew applications are not able to run natively on DSi due to being designed to run on a flashcard with DLDI, you should run homebrew from TWiLight Menu++ for the best compatibility.