When you begin playing a clip on YouTube, press the Pause button and wait a moment to see if the buffer continues to move on after a while (e.g. wait ~30 seconds). The following shows an example of the buffer:
Then press the "Play" button to resume playback. If the playback still stops where the buffer progress ends, then its likely a problem with your Internet connection, Wi-Fi (if using it) or your Internet connection is clogged. For example, if you use BitTorrent and simply leave completed Torrents seeding even without downloading anything, this alone is enough to disrupt playback on YouTube.
On the other hand, if the buffer continues to grow, but the playback does not continue, then it's likely a problem with your web browser or Flash plug-in.
Finally, if the buffer does not continue to grow with the playback paused (i.e. that red part does not continue to grow beyond the current position indicator), then something is interrupting your Internet connection. In this case, try downloading a large file (e.g. 15+MB) such as from here to see whether it completes successfully, as this will show whether it's just YouTube or if other transfers are affected.