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Is there accepted nomenclature for the star around which a particular exoplanet orbits? Meaning, if I were to say "The exoplanet blah blah blah's (noun)" what noun would I put there? Sun? Star? Designated Gravity Buddy? Something else?

As I understand it, this would not come up often currently in the real world (because Exoplanets are named by their star), and it more in the realm of science fiction and fantasy at the moment. However, just because something seems currently useless (to me) from a scientific standpoint doesn't mean that someone hasn't thought up a name.

Would this name change if you're actually on the planet or in the system? (ie, you hear about Tatooine having "two suns" not "two stars").

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Personally, I think "sun" would be a perfectly good term. However, these terms tend to have "official" definitions that aren't necessarily what you'd think. For example: there is technically only one moon in the Solar system, which is called the Moon, and all the similar objects around other bodies are called satellites. I can imagine there being something similar with the Sun being the only official sun, so I'll leave it to someone with more knowledge to give an answer. –  Nathaniel Jan 21 '13 at 3:23

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I think you answered your own question. The "star around which an exoplanet orbits" is called "star". We call our star the "Sun", or "Sol", and as far as I know, the term "solar system" is reserved for our star exclusively. Other systems would be called "stellar systems". Stars can have other names, and usually do. But only our own star is called "Sun" or "Sol".

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"Host star", or "host" for short, seems to fit the bill.

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