Word size is not a term originating in cryptography. Rather it is a term which came from the area of computer architecture. Here it specifies either how many bits are transferred over a bus at a time or the size of a register.
When used in the description of cryptographic primitives (such as a compression function or a block cipher), it covers the size of the variables in the definition of the primitive. Ideally these variable sizes line up with the register size of the CPU they are going to be executed on.
In case of the SHA-2 family, the internal state is always 8 words, and the block size is always 16 words. The word size is either 32 bits or 64 bits.
This is why on a 32 bit CPU SHA-224 and SHA-256 are faster than the other variants, but on a 64 bit CPU they are slower than the other variants.