Go to your phone's app store, search for YouTube, and click the install button on the YouTube app's page. For an iPhone, use the app store on iTunes to find the YouTube app; for an Android device, download the YouTube app from Google Play.
The YouTube app came preloaded on Apple iPhones until 2012; if your iPhone is newer than that, you must download the app and install it. The YouTube app is preloaded on many Android phones, such as the Samsung Galaxy S5. There is also a separate YouTube app designed for children to use; it only offers age-appropriate videos, channels and playlists.