The main difference between Gentoo and other Linux (Redhat, Mandrake, Slakware) distributions that I have used is the package management system, portage. To understand what portage does, I will give a brief introduction on how Linux distributions are structured.
The average Linux install is comprised of various parts. The kernel is the heart of the operating system. It is the software that runs between the hardware and the various applications the user runs. On top of the kernel are the basic system libraries and tools. On top of that are the applications that users interact with (Web browser, e-mail client, etc). Linux applications tend to be developed by small teams that focus on a single application. As the various applications (and libraries, tools and the kernel) are updated users want to get those updates on their systems. Getting changes to users is generally referred to as package management. A package refers to a particular application. Knowing the requirements of the package and how that relates to the state of the system is package management. To keep package management from being a problem for users, distributions supply software to deal with it.
Many distributions focus their package management around major releases. These releases contains particular versions of applications. As new versions of applications are developed the distribution adds the newer versions to the next release. This way all the little application updates get bundled to a big release of the distribution. This may work for some people, but I like to have my software updated as soon as possible.
Gentoo uses a piece of software called portage. Portage maintains a database of information about applications and their requirements, as well as what software is currently installed. A Gentoo user can tell portage to update its database (over the internet) to find out if new versions of installed software are available. If new versions are found, portage can automatically download and install the software on the computer. In addition, if one wants to install a new application, portage can handle that as well.
Furthermore, the user can configure portage to configure those packages in certain ways. For example, Linux has two major desktop projects, Gnome and KDE. A user can tell portage to use KDE or Gnome, or both. If an application supported both environments, portage would automatically configure the application to support whichever was selected (or both or neither).
In addition, new applications can also be installed via portage. As portage knows about all the requirement of particular applications, it can automatically determine what software should be installed to get that application to work. For me this is very useful. I like to try new applications. Previously if an application had a long list of requirements it could be quite difficult and annoying to get it installed. Now, if the application is in portage, I can install it by just typing a command.