It sounds like your CPU/GPU is not powerful enough to handle HD video. The graphics card in your images, a GeForce 6100. was released in 2005 and is old.
Flash does use the GPU to accelerate video decoding when available. So upgrading your GPU would seem to help and may help you playing HD video outside of Flash as well.
However, if you are using a 2005-era system, your CPU is likely a single core 2Ghz (EDIT: looks like I was wrong!) and you probably are working with 2Gbytes or RAM or less. You may have problems with HD video no matter what you do due to this.
One other thing you can try is an application called Minitube which will play Youtube videos without using Flash's video player (though I wasn't trying to play HD videos). I am able to play video on a 900Mhz Asus netbook with 1GBytes of RAM with little issue (on Linux, I might add).