IOC and Dependency injection have already been mentioned above, and I would urge you to look at them.
Largely, however, interfaces allow a contract to be specified for an object that doesn't require an inheritance model.
Lets say I have class Foo, that has functions x and y and property z, and I build my code around it.
If I discover a better way to do Foo, or another sort of Foo requires implementation, I can, of course, extend a base Foo class to FooA, FooB, MyFoo etc, however that would require that all Foos have the same core functionality, or, indeed that any future Foo creators have access to the base Foo class and understand its internal workings. In C#, that would mean future Foos could not inherit from anything else but Foo, as C# does not support multiple inheritance.
It would also require me to be aware of possible future states of Foo, and try not to inhibit them in my base Foo class.
Using an interface IFoo simply states the 'contract' that a class requires to work in my Foo framework, and I don't care what any future Foo classes may inherit from or look like internally, as long as they have fn x fn y and z. It makes a framework much more flexible and open to future additions.
If, however, Foo requires a large amount of core at its base to work that would not be applicable in a contract scenario, that is when you would favour inheritance.