Thats a BSOD (Blue Screen Of Death) which may possibly a hardware issue, although it does not explain why it boots correctly under "Normal" mode.

Reason being is in safe mode only the bare minimum of drivers are loaded to boot the system, so it would be the reverse if this was a problem, i.e. not booting normally.

There is no way anyone can even get an indication as to what this relates to without further specific information, at the bottom there is a line of text and also an error code which can be used to isolate to a specific error.
I have seen the same issue before re your web browsing on customer machine's before, it could be a result of virus etc giving bogus web server DNS entries for those sites.

Basically DNS is what resolves the domain name you know of, i.e. google.com to it's IP address, which is whats actually used for communication.

This is easy to fix IF you know what you are doing, otherwise it can be tricky to resolve, let me know if you have no luck and I will tell you.

Martin