It depends on what you're looking for. If you're a power user, you might like to try Gentoo Linux, which is a metadistribution that you can build to your exact specifications. It also has the benefit of being versionless once it is installed: simply doing #emerge -puND world will update the entire system. There is no need to ever install a "new version of Gentoo". Further, you can upgrade your system at your own leisure. The downside of Gentoo is that it is very time-consuming to install; the good thing is that once it is installed and configured, it requires very little maintenence, and never needs to be installed again (since it's versionless).
Another good choice is SuSE linux, which has a very good system called YaST for hardware management. Of all the distributions, I found that SuSE worked with hardware the best; it detected everything automatically, and set my user interface up to work with my hardware as I wanted to automatically as well, much as windows would have done. It is also very easy to install, and easy to use. It comes with a full Graphical User Interface (either gnome or KDE, although the default is KDE) installed, unlike Gentoo, on which one must install the GUI manually.
Fedora Core (or the paid version, Redhat) is a very old distribution, and has very wide software support. For example, Yahoo! Messenger is available in the *.rpm distribution form, which Redhat/Fedora Core uses. It is probably the most well-supported by companies with proprietary software, and is somewhat of a standard in the Linux world. If a company ever makes something for linux, they will make it compatible with Redhat, although they may or may not make it compatible with other Linux distributions.
Debian is another very popular distribution, and so are its offspring, Kubuntu and Ubuntu. Of the three, I would personally recommend Kubuntu or Ubuntu, since they are very easy to use, and use Debian as the base system. Debian has a very robust package management system (meaning software is widely available for the Debian distribution, and easy to install). Ubuntu is also known for its ease of use; if you want something that your grandmother could use, Ubuntu is probably your best bet. Kubuntu is exactly the same as Ubuntu, except that it uses KDE instead of Gnome for its Desktop Environment. Between KDE and Gnome, I prefer KDE because I like to configure my computer *just so*. For the "grandmother-capable" computer, however, I would recommend Gnome, as that project strives for simplicity and ease of use.
Breakdown of the distributions I would recommend:
Gentoo for extreme power-users and for servers, etc.
SuSE for a recovering Windows addict who actually knows his way around a computer, but is new to linux
RedHat for those who like to go with the standard (some people like vanilla just because it's safe and very well supported)
Kubuntu or Ubuntu for those who want the computer to Just Work (tm): Kubuntu for those who like to configure everything and click options in the "advanced" tabs, and Ubuntu for those who don't really want to mess with all that.