Carbon is Apple's previous application-development framework for Mac OS X for applications that needed to remain compatible with the Classic Mac OS.
Carbon is Apple's previous application-development framework for Mac OS X (macos), which existed primarily for applications that needed to remain compatible with the Classic Mac OS. Most of the original Carbon APIs existed both on Mac OS X and in a dynamic library (CarbonLib) for Mac OS 9.
Much of it is now deprecated and unavailable to 64-bit applications as of Mac OS X 10.6, but not all. Core Services (including the File Manager and even the Resource Manager) and much of Application Services (including core-graphics) are 64-bit-compatible, as are a few Carbon APIs, such as some parts of the Carbon Event Manager. Be sure of your ground before you tell a questioner they need to switch to Cocoa (cocoa); the API they're using may be perfectly safe, and may even be the only API specifically designed for their task (as part of Carbon Event Manager is for hotkeys).
Note that just because an API is in the Legacy Mac OS X Reference Library doesn't mean everything in it is deprecated or doomed. Again, Carbon Event Manager is an example.