Ubuntu is probably the most popular distribution amongst newbies. It is not, however, especially good for running on old computers without some serious tweaking.
Ubuntulite is a project for those who want to run Ubuntu on old hardware. It combines programs from Ubuntu's repositories with Ubuntulite's own repositories.
Fluxbuntu is a project with similar aims: Fluxbuntu should be lightweight, productive, agile, and efficient. It is based on Ubuntu and uses Fluxbox window manager by default.
Both of the distributions are certainly worth considering, if you want to install an Ubuntu-like system on an old computer.