Yum and apt-get are both package managers; they are programs that are aware of packages on an internet reposity (sort of "package warehouse") and can update their knowledge about what packages and versions there are. You use the program to download and install (or remove, update etc.) a package, and the program then calculates what packages are needed to install or upgrade to do the asked procedure, and after that download and install/upgrade (or maybe remove) the needed packages. So it's a semi-automated way to install and remove (=manage) software. In addition to internet reposities you can also usually define local reposities if you like, such as setup media that contains packages.
yum is the tool used by RedHat and Fedora. apt-get is the package manager of Debian. Basically they do just the same, except that the interfaces to those programs differ a bit, in addition to the more or less small differences in the ways the program work. Both can install, remove and upgrade software, and calculate dependencies and work with them. Debian's apt-get is usually used with Debian (and alike), and yum with RedHat/Fedora (and alike) operating systems, though you can probably use either one in your own distribution, as far as it's configured correctly. Often people say apt (apt-get is the command for apt as of now) is better than yum, but I don't know..personal taste. I wouldn't use apt-get on RedHat/Fedora-based systems, and I wouldn't use yum on Debian if it was possible.
The base package manager program for RedHat/Fedora like operating systems is rpm and for Debian (and alike) dpkg. Those can install, remove and upgrade packages that you have downloaded for them yourself, and calculate dependencies (tell you if the package can't be installed because it won't work before another package is installed), but they don't automatically fetch the pacakges for you - that's what you do yourself. So yum is like an "extension" to what rpm does, and apt-get for dpkg: they do the same things and more.
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