Firstly, what is this called? Is this the system's "platform"? If I want to know if a system is 32-bit or 64-bit, do I ask what "platform" it is?
Next, is what I wrote below correct:
-A 64-bit processor can run a 64-bit operating system or a 32-bit operating system (with a loss of efficiency).
-A 32-bit processor can run a 32-bit operating system only.
Assuming we mean on bare metal, the second statement is true. The verity of the first depends on the processor architectures. IA-64 processors cannot run IA-32 operating systems without emulation, whereas EM64T processors can.
I'd strike the part about "with a loss of efficiency". 64 bit processors can run 32 bit OS's just fine; with exception to Itanium which require special OS builds.
Efficiency has nothing at all to do with it.
To give a partial answer: 32-bit or 64-bit is part of the architecture. I guess it is part of the platform too, but you're more likely to make an expert think of the 32/64-bit distinction by talking about architecture.
Honestly, if you want to know whether a system is 32-bit or 64-bit, just ask, "32-bit or 64-bit?"
If you want to know if a system is 32-bit or 64-bit, you could ask what "bitness" it is.
Whether the statements about 64-bit and 32-bit processors running 64-bit and 32-bit operating systems are true depends entirely on which processors and operating systems you are talking about. You won't have much success running Windows XP on an 64-bit Alpha or SPARC processor.
If I want to know if a system is 32-bit or 64-bit, do I ask what "platform" it is?
"Platform" is an overloaded term that can mean a great many things. It can mean the CPU family: x86 platform, IA-64 platform, x86-64 platform, ARM platform, MIPS platform, SPARC platform, etc. It can mean the underlying operating system: Windows platform, Linux platform, Solaris platform, etc. It can mean a combination of these: Wintel platform (Windows + Intel). It can mean specific distributions: Debian platform, Slackware platform.
If you want to know if a system is 32-bit or 64-bit, ask if it's 32-bit or 64-bit. And make sure that you also check the CPU for compatibility for your purposes. ARM cores are 32-bit too, but you can't run Windows on them (at this time). SPARCs can be 64-bit, but you won't be running your copy of Microsoft Office on it, I'd wager.
A 64-bit processor can run a 64-bit operating system or a 32-bit operating system (with a loss of efficiency).
This depends very much on the processor. Intel's IA-64 chips can't run 32-bit operating systems because they don't really have 32-bit instructions (if memory serves). About the only way you could run a 32-bit OS on one is if you emulated a 32-bit CPU of some sort. This would suck performance-wise.
On the other hand the x86-64 chips can run 64-bit OSes or 32-bit OSes with no loss of performance at all for the latter (when compared to a pure x86, I mean). I'm running a 32-bit version of Ubuntu, for example, on an x86-64 chip without difficulties. Of course the 64-bit system will run faster than the 32-bit if the underlying software was written to take advantage of the expanded capabilities! (You'd be surprised how little it matters for most day-to-day tasks, though.)
A 32-bit processor can run a 32-bit operating system only.
Again, it all depends on the processor. An x86 (not x86-64) can run 32-bit OSes, but can also run 16-bit OSes right down to plain old MS-DOS. On the other hand, ARMs tend to be 32-bit only. (There are some ARM cores that have 16-bit instructions, but most do not, again if memory serves.)