I read Goldberg 2003, an overview of construction grammar. (In the paper they cite Goldberg's 1995 book).
A construction is a "pairing of form and meaning." Words, multiwords, morphological features, etc. are all constructions. To understand them you have to think about communicative functionality, meaning, and general cognitive constraints.
That's kind of it. Some examples:
It's a very NLP-friendly view of language. They're like the all the little features you get in MT or other tasks.
It's kind of like lexical semantics, except for more than just words.
It's kind of dissatisfying, in that there's no theory how words combine into meaning.
It's kind of not much of a theory. I don't see how it explains or makes testable hypotheses or predictions. Or even how you would use it to help design a grammar or NLP features, even. I don't see how you could hope to compare it to LFG or CCG or HPSG or minimalism or what have you.
I see how it might have helped Reichart and Rappoport think about these things, or something. I dunno if I should bother reading the book though.