Generally speaking, NLP is an engineering system whose main function is to effectively process natural language corpora in order to fathom out and portray the way human speech is produced and understood.As you have rightly observed, issues related to sentence context play a very pivotal role in simulating human speech processing by using machine level languages. For more information, I refer you to the following links, which can hopefully satisfy the targeted question.
I think, depending on the type of context you are looking at, you will end up in very different areas of research. You have a least, 3 different types of context for a word. NLP, to my knowledge, is at best looking at 2 of them:
1) word-context, meaning words surrounding your target word
2) supra-sentential context: what has been encoded in sentences prior to the one you're looking at but which may bear on the meaning of your target word (that's where the terms of Dhanji above are coming into play)
3) conversational context: This refers to stuff where the primary locus of language is seen as the person-to-person face-to-face conversation where stuff that I'm saying will sometimes refer to stuff that you're saying and vice versa. Also important here is what is not being said at all or might be hinted at by the strategic use of pauses and `filler' non-words. Other important issues here are common ground - what do I know that you know - which will determine how I shape my sentences and non-linguistic communication via gestures, body postures, etc..
As far as I know, only contexts 1) and 2) are touched upon by NLP. If you're looking at context 3, you're in the realm of pragmatics and conversation analysis, the former of which kind of is done by linguists, whereas the latter comes from sociology.
The methods there are very different from the one employed by NLP.