Dependencies are any package that need to be installed first so that you can install the package you want. So if you want to install Xorg to have a graphical interface, you would need to install many different libraries first. Or if you wanted to install KDE, you would first need to install X (of some sort), kdelibs, and many other packages.
Some modern distros handle this for you, so you don't really see it going on. Ubuntu/Debian has apt-get, Fedora has yum, Gentoo has emerge, etc.
Hopefully after you use Ubuntu for a while and become confortable, you'll be interested in learning more about Linux. May I suggest you check out Gentoo then? If you like to be in control and have choices, then that is the distro for you! I cut my teeth on Red Hat, but jumped to Gentoo a month after that. Sure, it was tough, but I learned a LOT really fast. And if you have problems, they have a great community of users to help you out.