What does it mean in layman's terms? Will I be able to run a Carbon application under Leopard, or is it time to start looking for an alternative?
Carbon and Cocoa are two different sets of programming tools for programmers in OS X. There're a number of threads that debate this, and it's a can of worms, but to oversimplify, Carbon is basically more like what programming looked like on Macs before OS X, and so was supposed to facilitate developers moving over from OS X. In contrast, Cocoa was sort of the gold standard of what the development environment was supposed to look like in OS X.
Both Cocoa and Carbon work on both PPC and Intel. They're both going to work in Leopard. There are two important issues, though:
1) There are a few big apps or suites (e.g. MS Office, Adobe CS) that are written using Carbon. What this means is that these apps take a long time and a lot of money to port and convert, and so to the extent that they can benefit from 64-bit, it's important that they may not be able to get it unless they switch to Cocoa (Office probably wouldn't benefit, but CS might).
2) There are some small system level tools that only work in the Cocoa environment. For instance, Cocoa provides a built-in inline spell checker for the text areas in apps, but this isn't something that Carbon apps can use. So this is why it took such a long time for built-in spell checking to happen in Firefox (which is not Cocoa right now, I think, but will use Cocoa later, I believe in version 3). I think another example is the Ctr-Opt-D command that triggers the OED pop-up. Things like international language support also don't work completely in Carbon, which I think is why some languages don't function correctly or behave bizarrely in Office.
Anyway, the second issue has nothing to do with 64-bit. It has to do with user interface niceties. The first issue does have to do with 64-bit, and really only applies to apps that can benefit from 64-bit. The rule of thumb is that an app needs to push around 4GB or more of information to really benefit. So, think some photo / art / video / rendering apps, some kinds of scientific computation, etc. This is why iTunes becoming 64-bit is nonsense.