<p>Inversion of Control (IoC) can be quite confusing when it is first encountered.</p>

<ol>
<li>What is it?</li>
<li>Which problem does it solve?</li>
<li>When is it appropriate to use and when not?</li>
</ol>

Wikipedia Article
. To me, inversion of control is turning your sequentially written code and turning it into an delegation structure. Instead of your program explicitly controlling everything, your program sets up a class or library with certain functions to be called when certain things happen.
It solves code duplication. For example, in the old days you would manually write your own event loop, polling the system libraries for new events. Nowadays, most modern APIs you simply tell the system libraries what events you're interested in, and it will let you know when they happen.
Inversion of control is a practical way to reduce code duplication, and if you find yourself copying an entire method and only changing a small piece of the code, you can consider tackling it with inversion of control. Inversion of control is made easy in many languages through the concept of delegates, interfaces, or even raw function pointers.
It is not appropriate to use in all cases, because the flow of a program can be harder to follow when written this way. It's a useful way to design methods when writing a library that will be reused, but it should be used sparingly in the core of your own program unless it really solves a code duplication problem.