You would use video editing software such as Sony Vegas, Adobe Premiere, Final Cut, etc. All of these cost hundreds of dollars but some of them have free trials.
If you just need something free and simple, most people on Youtube get by with Windows Movie Maker (PC obviously) or iMovie (OSX). The results are far from astounding but if you just need something to clip pieces together the free options are the best choice. If you need anything beyond that, I am sure there are some decent and cheap consumer grade video editors out there but unfortunately I haven't used any to give a recommendation.