Video editing software is a lot more complex than audio software. The fact is that there are few if any viable FOSS alternatives to commercial software.
Cinelerra is a full-featured editing system designed to replace expensive professional systems. This project has been on and off again for several years. They’re promising a big new build in March 2015, so it’s one to watch.
Avidemux is a simple video editor and encoder. It cannot be called an “editing system,” but for many small chores it is extremely versatile.
If you’re on the absolute bleeding edge, you may want to watch what’s going on at Saya-VE. They’re building a FOSS editor, and haven’t even got the binaries together yet. Exciting!
Mac users have another options that are extremely useful – iMovie. Whereas not a full-featured professional editing system, iMovie has more than enough functionality to cut audio and video for your projects. Windows used to ave the ungainly and quality-destroying MovieMaker, but that’s no longer an option. Windows no longer has a built-in video editing program! Maybe that’s just as well, because MovieMaker was awful.
In addition, these video tools may be useful to you:
•The venerable VLC is everyone’s favorite cross-platform media player.
• TMPGenc is a popular video encoder for Windows and Linux. It encodes AVI files to MPEG-2 for DVD at significantly higher quality than many authoring programs.
•Mencoder and MPlayer are venerable open-source utilities that many other FOSS programs require or reference.