Summary: Video editing is storytelling. There are many programs and tools that will help you assemble your story in an effective and entertaining way.
The whole reason to go digital with video is the ease in which we can manipulate the media we use. With our editing program, we can literally drag a section of the video footage from one area and drop it into another area and see the results instantly. I should say that we can at least preview it instantly. More on that later. We can also use the editing program to apply visual effects like different transitions between scenes, or color filters, or even add titles.
The premium video editing programs that are used today are Adobe Premiere (available on the Mac and Windows) and Final Cut Pro X (Macs only). On the free side, Windows computers come with a program called Movie Maker, and Macintosh computers come with iMovie. Both are basic editing programs, but iMovie (version 10) is very much like Final Cut Pro X, but with a reduced feature set. There are still around four versions of Movie Maker depending on what version of Windows you are using.
Using these editing programs is pretty straight forward. You drag video clips to an area of the program called a timeline. You can then add transitions and titles, as well as music, in a separate audio track. The programs are generally intuitive and you can edit your video project quickly. However, you should save some time torender the video. What is rendering? Well, all of the editing you are doing is kind of a miniature preview of the final project. When you save your final movie (or upload it to sites like YouTube or Vimeo), you must sit back and let the computer create all of the frames of video including the transitions, titles, etc. The more special effects and changes to the original video you have, the longer the rendering time generally is.
Once your video is finished, your output is limited only by the ability of your editing software. Generally your options would include, uploading to a video sharing site such as YouTube or Vimeo, creating a video appropriately sized for a mobile device, making a DVD (a less popular option these days), or just creating a video that will play on a computer. In all of these cases the video editing program will encode it into a specific format.
Wikipedia has a pretty extensive list of video editing software that is available.