Normally in English dependency means something like dependent or subordinate: something that depends on (relies on) or is subordinate to something else. My children can be called my dependencies because they depend on me; without me, they have no legal status. A country's dependencies are states or other bodies subordinate to it because they are not strong enough to be independent. And so on.
In the world of computer programming it has the opposite meaning: an additional program or script on which the program in question depends. Here are a couple of examples:
Some plugins and CSS components depend on other plugins. If you include plugins individually, make sure to check for these dependencies in the docs. (Documentation for Bootstrap, accessed 20150517.)
(The "other plugin" is a dependency of the main plugin being discussed; the main plugin depends on the dependency.)
A dependency occurs when one package depends on another. … These dependencies really work in your favor, since a security bug fix in the network libraries can update all applications that make use of the updated libraries. (Documentation for Fedora, accessed 20150517.)
(Many applications may depend on a single dependency; fixing that dependency helps all the applications that depend on it.)
This is the opposite of the traditional use of the word. Here, the main program or script I am running relies on a "dependency", rather than the dependency relying on the main script I am running.