When I try to watch a video with a high HD Resolution 1280 x 720 H264 in Ubuntu it isn't played smoothly. Stops and glitches occur. I don't think this is a problem with the CPU or graphics card, because under Windows there are no problems like this. Could it be a driver problem?
Update: For me VLC Player works best. No stops and no glitches. Smplayer also looks good but the performance leaks similarly to the standard Ubuntu video player in my case.
You can't compare Windows and Linux, because video driver performance and features are different. For instance, if you are using nVidia you need a card that supports vdpau for best performance, which is equivalent to PureVideo in Windows. PureVideo supports many more cards. However, depending on your CPU, you can play HD videos smoothly without vdpau support. So it depends on your video driver, CPU and also player settings.
It would help if you could provide more info about your video card and CPU.
Anyway, try smplayer
It's the best player IMO and plays even videos with resolutions higher than 1080p on my Core2 Duo with nVidia 7300 GT (not vdpau).
Make sure you tweak smplayer. In the Preferences, click the Performance section and select the option Skip only on HD videos in the Loop filter settings. In the General section, click the Video tab and make sure the Output driver is set to xv.