Munny is the name of the currency (possibly universal currency) both found and used in the Kingdom Hearts universe.
Its name appears to be an intentional misspelling of the word 'Money' in the way that 'Honey' is spelled 'Hunny' on honey jars in the Winnie the Pooh stories. As Hundred Acre Wood - home to Pooh and friends - is an explorable world in the Kingdom Hearts universe, the spelling of 'Munny' is quite possibly meant to reflect the spelling of 'Hunny'.
Gaining Munny
Munny may be obtained in all the Kingdom Hearts games (except for Chain of Memories - please refer to Moogle Points for more information) by several different means.
The most primary way of obtaining munny is by defeating the Heartless or Nobodies. In this way, munny can be acquired through Munny orbs - orbs that are half-gold and half-blue - which come in small, medium, and large sizes with the smallest worth 1 munny, the medium-sized worth 5 munny, and the largest worth 20 munny.
A second way to acquire munny is by using Trinity Marks; Trinity Marks tend to yield many orbs.
A third way to get munny is by selling goods and synthesis items.
A final way to get munny is by earning them through mini games or side quests. For example, in Kingdom Hearts II, when playing as Roxas, you have to complete various mini games and errands to earn enough munny to take a train trip with Hayner, Pence, and Olette to the beach before the summer ends and school begins.
Losing Munny
Typically, the major way to 'lose' munny is by purchasing something and thus spending the munny.
The only other way to lose money is if Sora slips on a banana peel thrown by Bouncywilds in the Deep Jungle World. Once he slips, a portion of his money will fly everywhere, forcing the player to hurry and recollect it all or lose it. On the other hand, Bouncywilds can also slip on them, dropping large quantities of Munny.
In Kingdom Hearts II, Roxas loses some hard-earned munny to a thief.
Munny as a Plot Device
Munny is used as plot device in Kingdom Hearts II when Roxas (along with his friends) in Simulated Twilight Town earns munny by doing odd jobs to fund a trip to the beach only to have the munny pouch containing all the munny stolen from him by Riku in the form of Xehanort's Heartless as the beach didn't exist in the Simulated Twilight Town.
This munny pouch filled with Roxas' hard-earned munny is later given to Sora.