For instance, I would like to extract a screenshot or page only containing the summary of the featured article of wikipedia (http://en.wikipedia.org/wiki/Main_Page).
It should not be of the whole page.
Can you be a little more specific? The problem is that the definition of "part of a page" is completely arbitrary, and it will vary from website to website. So any solution that purports to do this will be limited to some finite subset of all websites out there.
Are you looking to not even retrieve the other bits of the page at all (on your client) or are you wanting to retrieve them and then discard them? You could do something like this clientside quite easily. Getting it through a web service somewhere would be equivalent to going through a filtering proxy, and many of the technologies used for filtering proxies (e.g. privoxy) could be used there, but you'd have to write your own code to determine what's relevant and what's not.
There are too many different possible ways to architect a webpage to reliably create something like this that will trim the "chrome" off of any webpage and only show you the content. For every solution you offer me, I could craft a webpage that would make your solution fall down.
If you want to know why this isn't possible in the general case, then consider this problem: design me a machine that, without any human intervention aside from physically moving the device within reasonable distance, will perfectly prune any tree, of any height or width or species, to an appropriate size. Good luck with that.