a whole nother day

What is the meaning of this? Is it grammatical?

Yes -- at least in spoken US English. Perhaps it could be written as:

"a whole 'nother day." (although you never hear "a whole another day")

Also
"A whole other day"

It means "a completey different, new, or other day"

The motto of Texas is "It's like a whole nother country."

Let's not forget "A nickname" which is an incorrect division of "An Eke-name" ('eke' = also, alternate)

there are many other examples too