This is usually a problem with your connection (maybe a firewall on your WiFi?). But it can also mean a problem with the app in very rare occasions. Here's what you could try:
1) Try opening the YouTube video you want in a web browser. Does it play? If so, then the problem lies within the app. If it doesn't, the problem is with your Internet service provider (or carrier, if you're using mobile data).
2) If the problem is with the app, you can try going into the Application Manager, search for the YouTube app, and try clearing Cache. If this doesn't work, try clearing Data as well (just remember that this means you'll have to log in again if you have a linked account). If this still doesn't resolve the issue, you can try uninstalling the app, rebooting, and re-installing.