Actual Gesture: Covering the mouth with the tips the fingers, whether it's the front, back in a cup shape
It's Function?: To hide your teeth and cover up while you hold back the chuckles
Any Associated Phrase?: No, but the implied one is “Sorry I find your current state hilarious”
Anything else?: It's a rather formal-and perhaps feminine?-gesture, and back in the day, flashing the choppers meant either showing the person in front of you a crack in your armor or that you weren't that serious of a person, either implying poor form on your part; in addition to that, women had to deal with the fact that pearly whites meant you weren't hitched yet-and possibly never would be, since married women had theirs painted black, the practice known as 御歯黒「おはぐろ」. These days, though, it's more likely they think they got a bad case of the yuck-mouth, and would prefer you not see them on display.
Literal Meaning: There is business
Intent: I got a thing do to
Usage: Much like the phrase "I got a thing to do", it tells that there's something else that needs our attention without specifying what it is, thus insuring that your private and public lives stay separate.
Notes: This phrase also has few interesting variants to it, like 用がない(it and any variant of it saying you either don't have anything to do or that you have no need/use for something) and 用を
(So you 'got something to do,' eh? I get you, man. See you 'round)
(If it's not useful to you, why not just throw it out?)
(Pardon the interruption. Off to go take care of business)