The International Version is an updated re-release of a game that originated in Japan. The term comes from it being based on the version released outside of Japan. Sometimes this version will contain new content on top of what the Japanese and Western versions have and/or can contain multiple language voice over tracks.
Squaresoft (now called Square Enix) is one of the major companies to do International Versions doing them for many of their major games and even some spin-offs. Square will have different reasons for releasing an International Version, depending on the game. In all cases where the games contain voiceovers, the International Version will contain English voiceovers (only some Square International Versions will contain Japanese as well). Here are some of the major cases:
Dirge of Cerberus & Dissidia
Dirge of Cerberus: Final Fantasy VII for the Playstation 2 and Dissidia: Final Fantasy for the PSP received negative reviews upon their original release in Japan. When those games got released in Western territories, they received major overhauls to their gameplay in order to make the games more enjoyable. Square then released direct ports of the Western versions of the games in Japan to better praise.
Final Fantasy VII & XIII
Final Fantasy VII was a huge success when it was released in Japan, but Square took the opportunity of the Western localization to expand upon the storyline and altered minor aspects of the gameplay. The International Version released only in Japan, is a direct port of Western version. As PS1 Final Fantasy games did not contain voiceovers, it is the only International Version from Square not to contain any English elements.
Final Fantasy XIII was released for the Playstation 3 only in Japan, but in the West, the game marked the first time the series came to Xbox 360 as well as Playstation 3. The Xbox 360 version later came to Japan as the International Version. There were only minor tweaks to the gameplay done between the various versions, the biggest change to the International Version, aside from the use of English voiceovers, was the change from the Japanese theme song to the English theme song.
Final Fantasy X, X-2, & XII
The Japanese and North American releases of Final Fantasy X are for the most part the same. Being a major release for Square, an International Version was later released adding several pieces of new content. Interestingly, the European release of the game was released after the International Version, so the European version contained the content from the International release.
The original Japanese version of Final Fantasy X-2 is the same version that got released in North America and Europe. An International Version was then released containing new content and a brand new side-campaign.
Originally, Final Fantasy X's International Version and Final Fantasy X-2's International Version were not available in certain regions, but recently Final Fantasy X/X-2 HD Remaster was announced for Playstation 3 which will bring both International Versions to all regions.
Like X-2, Final Fantasy XII is for the most part the same between the original Japanese version, the North American version, and the European version. When the game got an International Version, the game received a major overhaul to its systems, to the degree that it is almost a brand new game. The International Version of XII remains exclusive to Japan.
The original Kingdom Hearts for the Playstation 2 and Kingdom Hearts: Birth by Sleep for the PSP received new content when they were released in the west. Not only did the International Versions of these games, titled Final Mix, contain the content from from the western releases but added more new content on top of that.
Kingdom Hearts II, on the other hand, is content identical between the three regions, barring the fact that certain content in North American version had to be altered or even censored to get an E10 rating from the ESRB. When the game was given a Final Mix, altered content was changed back to its original state and added a ton of new content exclusive to this version.
All three Final Mix versions were Japan exclusives upon their original release on the PS2 and PSP. However, Kingdom Hearts Final Mix was later released worldwide as part of Kingdom Hearts HD 1.5 Remix for Playstation 3 and Kingdom Hearts II Final Mix & Kingdom Hearts: Birth by Sleep Final Mix will available worldwide as part of Kingdom Hearts HD 2.5 Remix also for Playstation 3.
With the release of Final Fantasy XIII-2, Square has appears to have ended the trend of doing International Versions for their games, opting instead to add new content to their games via the more modern approach of Downloadable Content.