What is Component-Driven Development?

What is it?

I think the definition in your answer covers this question well. Although, I question why the definition includes that a component needs to explicitly define its dependencies. A canonical example of a component is an ActiveX control - do they need to explicitly define their dependencies?

What problems does it solve?

Management of complexity. It seeks to address that by allowing you to only ever think about the implementation of the component. One should only need to author components, one should not to have to think about how to combine or manage them. That is done by some framework or infrastructure external to the component, and unimportant to the component author.

When is it appropriate and when not?

Not necessarily appropriate in a trival or throw-away application. The bad smell in a component architecture, is if you are spending time on thinking or working on the infrastructure to manage and combine components, rather than the components themselves.

I am not sure it is a "widespread" terminology, but in VCS (Version Control System), I know of two ways to manage a set of files needed to build a program:

• system-based approach, where the all set has a common life cycle and must be tagged as a all
• component-based approach, where individual set of files have their own life cycle, and where a meta-label references all the labels of the components to designate the all system by composition and dependencies between those components.

The applicative architecture is used to identify those components:

• functional domain and applications
• third party libraries
• frameworks

That is where IoC comes in, since it is at the base of any framework. The problem it solves is allow you to better identify the part of your application:
Suppose you design a PLR (Profit and Loss) application, in charge to compute the gain and losses (position) of a trader.
You would quickly realize it is not a single application, but a composition of several:

• GUI
• launcher
• dispatcher (to dispatch the computation across several server, because one would not have enough memory to compute all!)
• and so forth

You can then identify a computation framework (Ioc) which would enable you to plug-in your different modules, which then are called at the right time by your framework.

Or you can identify purely technical frameworks (KPI, logs, exception managements) which can then be used by any of your other functional components.

In term of project management, that also allows you to develop each part independently, while assuring a global coordination through the VCS.