Photo/Video Editing or Management
For an open source alternative to Adobe Illustrator, you can try Inkscape. It's a vector graphics editor similar to Illustrator, CorelDraw, or Xara X.
If you want to work with photos, GIMP is similar to Adobe PhotoShop.
Looking for a simple image editor for Linux, like Paint in Windows? Give Tux Paint a try.
For simpler editing, or for one that also runs on Windows, try Avidemux. It still supports editing AVI, DVD-compatible MPEG files, MP4, and ASF using a variety of codecs. DVD authoring is supported using dvdauthor.