Anyone using any of the 'IX' operating systems will be familiar with Open Source software. For video editing there are several packages avialable. Here is a brief overview:

Cinelerra

One of the first Linux based real-time editing and special effects systems. Having been around for some time the package is relatively bug free and feature rich.  It really offers the full package, capturing, compositioning and producing video with high quality audio. That's all on the plus side. The down side is it is fairly complicated for the average user. Many would get bogged down with all the features on offer. If you're making a feature film or documentary, check it out. If you're more interested in combining baby's first steps with a backing track... it probably isn't the package for you.

Visit the Cinelerra website here.

Avidemux

Avidemux is not strictly for Linux as it is available for various other platforms. It's a small package with simple features. Essentially, it is for basic cutting and editing and combining of footage. My experience of it was not good, it simply didn't provide enough features for me. However, it does come with many codecs so it can be used to edit most video files. For simple tasks it may be what you need. If you have a different experience of it, let us know.

Find Avidemux here.

LiVES

Lives is a Video Editing System (clever isn't it). Well I thought so! Unfortunately, I haven't used LiVES so my practical knowledge of it is limited. Flexibility is probably its biggest plus, as well as being easy to use from the start. Have a read and give it a try, then tell us what you think.

LiVES

This is by no means a comprehensive list. Additions and comments are appreciated, wanted and applauded.