In most cases you don't need a separate application (app) to use YouTube. However, there is an app available for Nokia/Symbian, Windows Mobile and Sony Ericsson devices.
You can view videos from your compatible phone by going to m.youtube.com.
If your phone can record video, you can upload video by emailing them from your phone. Log in to your YouTube account and visit www.youtube.com/mobile to find out your secret email address.