An increasingly fashionable word in Web development circles is 'Framework'. "Wow this system, theme, whatever uses wizzygeek framework." If you're a WordPress enthusiast I suspect you'll know what I'm referring to.
First fact, the idea of a framework is not new, it predates me getting into web development, which makes it old. Here's what Wikipedia say about it. To put it simply a framework can take many forms. It's primary purpose is to provide all the bits you need so that one can concentrate on developing the functionality of the software. If you buy a screw from a hardware store (they do exist somewhere) it comes with a thread so that it will go into the wood when turned and a slot in the top so that a screw driver can turn it. That in software terms is a framework. No one expects you to give the screw a thread or to have to cut the slot in the top.
How much impact does or should a framework have on the final result when creating websites? This is down to personal preference as some frameworks do a great deal more than others. We specialise in bespoke websites so we like a framework to take some of the slog out of development without imposing it's own look on the site. With this in mind our Framework of choice is Bootstrap, which happens to be the same Framework Twitter uses. It was developed by the Twitter developers as an Open Source framework. If a framework imposes to many pre made rules then you can spend as much time undoing those as you might have done not using that framework in the first place.
So don't be mislead by Themes with fancy framework names. Find what works for you.
Hope that clarified the matter for you a little.