What are 16, 32 and 64-bit architectures?
What do 16-bit, 32-bit and 64-bit architectures mean in case of Microprocessors and/or Operating Systems?
In case of Microprocessors, does it mean maximum size of General Purpose Registers or size of Integer or number of Address-lines or number of Data Bus lines or what?
What do we mean by saying "DOS is a 16-bit OS", "Windows in a 32-bit OS", etc...?
Answer
My original answer is below, if you want to understand the comments.
New Answer
As you say, there are a variety of measures. Luckily for many CPUs a lot of the measures are the same, so there is no confusion. Let's look at some data (Sorry for image upload, I couldn't see a good way to do a table in markdown).
As you can see, many columns are good candidates. However, I would argue that the size of the general purpose registers (green) is the most commonly understood answer.
When a processor is very varied in size for different registers, it will often be described in more detail, eg the Motorola 68k being described as a 16/32bit chip.
Others have argued it is the instruction bus width (yellow) which also matches in the table. However, in today's world of pipelining I would argue this is a much less relevant measure for most applications than the size of the general purpose registers.
Original answer
Different people can mean different things, because as you say there are several measures. So for example someone talking about memory addressing might mean something different to someone talking about integer arithmetic. However, I'll try and define what i think is the common understanding.
My take is that for a CPU it means "The size of the typical register used for standard operations" or "the size of the data bus" (the two are normally equivalent).
I justify this with the following logic. The Z80 has an 8bit accumulator and an 8 bit databus, while having 16bit memory addressing registers (IX, IY, SP, PC), and a 16bit memory address bus. And the Z80 is called an 8bit microprocessor. This means people must normally mean the main integer arithmetic size, or databus size, not the memory addressing size.
It is not the size of instructions, as the Z80 (again) had 1,2 and 3 byte instructions, though of course the multi-byte were read in multiple reads. In the other direction, the 8086 is a 16bit microprocessor and can read 8 or 16bit instructions. So I would have to disagree with the answers that say it is instruction size.
For Operating systems, I would define it as "the code is compiled to run on a CPU of that size", so a 32bit OS has code compiled to run on a 32 bit CPU (as per the definition above).