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.