There are many flavours of Linux, these flavours are known as distributions or ‘distros’. My favourite distro is Ubuntu although I have tried a few over the last ten years.
I have put Ubuntu on a few netbooks, but for those who are new to this sort of thing there are some distros especially for netbooks. For the EeePc you might like to try EasyPeasy (get it? Eeesy Pc) which has been made to be very easy to install. Jolicloud is really easy to install too butg it has some quirks. Linux Netbooks is a site especially to help netbook user with Linux and you will find an introduction to even more distros there.
For most netbooks my preference would be to use the normal desktop/laptop distro of Ubuntu, even though Ubuntu do a special netbook remix. If the netbook has very little power or memory the OpenOffice suite (like Microsoft Word) can be replaced with Abiword, an excellent, lightweight word processor.
But could the new kid on the block be a future winner? MeeGo is the result of Intel and Nokia working together to produce a lightweight Linux distro that will work not only on netbooks but on small smart phones too. With these two giant companies behind MeeGo, it is likely to make a big splash.