APT and RPM
The aptly named APT (excuse the pun), is truly a marvel: software is organized into categories, and each software package has certain meta-information attached, and more importantly, on what other software packages it depends. Software package lists are called repositories, and are in essence directories of packages organized in a specific fashion.APT uses this information in order to cross-check all dependencies on the system and make sure nothing is missing, and all this is handled in a manner transparent to the user.
There is another packaging system that tries to do the same, but with varying degrees of success: RPM (Redhat Package Manager).
RPM is used by RedHat and SuSE based distributions, APT is used by Debian and Ubuntu based distributions (though Ubuntu is Debian based, it has spawned itself many other distributions). The main difference between the two is how they solve dependency problems, when and if they arise.
APT is smart enough in most cases to inform the user and provide a way out of the problem. Even with multiple software repositories (listings of software packages), APT manages to keep different versions of the same software in the same system without interfering. In worst case scenarios, removing an offending package will not break the system and make other software unusable.
RPM has a more troubled background. RPM is very sensitive to versions, and not very forgiving of bad packages and user mistakes. The commonly used term 'RPM hell' refers to a situation where RPM fails to find or propose a solution for solving a dependency, usually prompting the user to remove all interrelated software, and breaking the system in the process.
That is mainly because the dependency solving is not built-in the RPM system and utilities, but handled differently by each distro.