What is a dependency property?

The only explanation I found helpful and well written is this one: http://www.wpftutorial.net/dependencyproperties.html

Basically, DependencyProperties differ from regular properties in that they're not just setters / getters for fields in the class, but they retrieve their actual values dynamically during runtime. The SetValue() method of DPs is pretty straightforward and sets the local value of the property to the value you gave it. However, when you try to GetValue() from a DependencyProperty, it will first look for a local value, if none is present (which is viable in DependencyProperties unlike regular properties) it will continue up the logical UI tree until it will find such value. If the framework has reached the top of the tree without finding any local values, it will then use a predefined default value as the property's value.

This method allows DependencyProperties to consume less memory than regular properties since only values that were explicitly set by the user will be stored locally.

And, as mentioned above, DependencyProperties also allow us to bind to them in the XAML code and set triggers on them, which on regular properties is not allowed.

I hope I've managed to clear some of the vagueness :)

Dependency properties are properties of classes that derive from DependencyObject, and they're special in that rather than simply using a backing field to store their value, they use some helper methods on DependencyObject.

The nicest thing about them is that they have all the plumbing for data binding built in. If you bind something to them, they'll notify it when they change.