which ubuntu installation will be the best for my PC
 

I have:
Motherboard - P5P800
RAM - 2x1GB and 2x512MB
Processor - Intel 2,8GHz dual core
HDD - 2x320GB
DVD player - Asus
DVD Writer - LG

I had tried to install 1.7 Ubuntu UE and 8.04 Alpha 6 but there is some problem - I can not finish installation. Only installation that is working is ubuntu live from CD. Anybody have solution?
Thank you in advance.

Alpha 6 is in the alpha stage (i.e. not stable), so it's not a wonder if it won't install. I installed it yesterday, first try freezed at 15% (first launched the live-cd, then started installer from the desktop icon). Second time I took the "install Ubuntu" option right from the boot menu and it went trough it without trouble -- but before trying to install, make sure that the disc is ok (there's an option to check the disc in the boot menu).

Any one of the Ubuntu versions would run on that machine, except for alpha/beta versions and such. The latest stable one is 7.04 which is available at the Ubuntu site - use that. In the end of April the new Ubuntu version should become stable, and then you can upgrade (I recommend a clean reinstallation, not upgrading trough package manager, even though that is possible too). If the Desktop disc installation hangs up, try to download the Alternate Install disc; it allows you to install in text-mode (don't worry, it's just as easy), bypassing some trouble that may come when using the live-cd - after the setup the system is just the same as it would be if you installed using the Desktop (live-)cd, so it's all the same.

Doesn't the live CD have an option to install to HD? Just do that.