I can't remember how to see if my OS X (10.6) installation is running in 32-bit or 64-bit mode.
Go into System Profiler and select Software from the left pane. In the right pane you will see 64-bit Kernel and Extensions. If it says yes, you are in 64-bit mode, if not, you're in 32-bit mode.
OS X doesn't really have overall "modes" as the question assumes -- various components (i.e. processes and the kernel) can independently run in different modes. Basically, whenever OS X goes to launch some piece of software, it looks at what architectures the software supports, what hardware you're running on, and tries to make the "best" choice about what mode to run that particular piece of software in (Macintosh Performance Guide has a good set of notes on how to override this).
If you want to know what mode the kernel's running in, John T's answer is correct (for 10.6; earlier versions only had 32-bit kernels). Note that while 10.6 is capable of running the kernel in 64-bit on many Mac models, it only actually does so by default on recent Mac Pro and Xserves, and AIUI even then only under the server version of OS X.
If, on the other hand, you want to know if OS X can run applications in 64-bit mode, then Arjan's answer is correct for both 10.5 and 10.6 (but not 10.4, which could only run pure-posix programs in 64-bit mode). Note that this is primarily controlled by the computer's CPU: Core Solo and Core Duos are 32-bit only, while Core 2 Duos and Xeons are 64-bit capable. There is no problem at all running apps in 64-bit mode under a 32-bit kernel (or vice versa).
Finally, if you want to know whether a particular program is actually running in 64-bit mode, Activity Monitor will tell you under both 10.5 and 10.6 (I dunno about 10.4).
I'll take this opportunity to recommend Prince McLean's 'Road to Mac OS X Snow Leopard' series at AppleInsider: '64 bits', '64-bits, Santa Rosa and the great PC swindle', 'Twice the RAM, half the price, 64-bits', and 'The future of 64-bit apps'. They were written before 10.6 was released, but still do a good job of covering the implications of various combinations of 32- and 64-bit mode operation.
For 10.6, John T's answer is right. Just for whoever gets here with the same question for 10.5: find the program Chess in Finder, right-click, Get Info. You'll see an option to fall back to 32 bit, if 10.5 is capable of running in 64 bit mode:
(This also works for non-Intel hardware.)