My laptop has 1GB of RAM and an Intel Pentium processor. Which distribution would be best for it?
Specs are here.
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Welcome in Ubuntu Community :) Because your laptop can have only up to 1 GB memory I suggest you to go with Ubuntu 12.04 LTS (Long Time Support) 32 bit or 12.10 32 bit. Is should work out of the box. You can download it here: http://www.ubuntu.com/download/desktop.
It's important that you will install 32 bit version. You can find some discussion about 32 and 64 bit architecture here.
With just 1Gb of Ram I would go with Lubuntu or maybe Xubuntu.
However it seems you have a display driver problem. Try a liveDVD or liveUSB to make sure everything works before installing.