What makes a CPU be called for example: "this CPU is n bits"?
Short answer: There is no universally accepted definition.
Less short answer: If the CPU supports all of the basic primitives on a 16-bit datatype, then it would probably be considered sixteen bits by the majority of users.
This has been a "holy war" since 1976 or so, and there is no "right" or "wrong" answer. Was the 8088 sixteen bits? Probably. So, was the Z80? It had some sixteen bit math, and an 8-bit databus. (Probably not -- the Z80 had no native 16-bit logical instructions, only add and subtract).
The question surged again when the 68000 with its 32-bit registers, and rich set of 32-bit operations appeared, but an internal 16-bit ALU and an external 16-bit databus (and then just to throw MORE confusion, the 68008 variant, with an 8-bit databus).
The bitness of a CPU is the width of the word it can process natively as a whole. This is generally the width of the registers and the ALU. For example, PIC 10, 12, 14, 16, 17, and 18 are all 8 bit processors. PIC 24, 30, and 33 are 16 bit processors, and PIC 32 are 32 bit processors.
Note that ALU and register width isn't the only thing commonly specified in bits for a processor. The instruction word width, or at least the width of the instruction data bus is another measure. This doesn't need to be the same as the ALU width, and often isn't. The same "8 bit" PICs listed above have different instruction widths. Sometimes you hear this referred to as the "core" width. For example, the original PIC 10 and 12 were 12 bit core machines, the mainstream PIC 16 has a 14 bit core, and the PIC 18 a 16 bit core, despite each of these being "8 bit" processors.