Learn Croatian: Kakva je današnja prognoza?

It is derived from the adverb danas (today). To turn time adverbs into adjectives, Croatian frequently uses the suffix -šnji or -nji. Here, danas combined with the suffix creates the base današnji. Because it modifies the feminine noun prognoza, it takes the feminine ending -a, resulting in današnja.

Why do we use Kakva instead of a literal word for "what" like Što?

Kakva translates literally to "what kind of". In Croatian, you use kakav, kakva, or kakvo when asking for a description or the characteristics of something. If you used Što here, it would sound like you are asking for the dictionary definition of the word "forecast" rather than asking what the weather will actually be like.

Does današnja decline like any standard adjective?

Mostly yes, but because its stem ends in the soft consonant nj, it strictly follows the soft (definite) adjective declension. This is important to remember for other genders and cases: the neuter nominative will end in -e instead of -o (like današnje vrijeme), and the masculine or neuter genitive takes -eg(a) instead of -og(a).

Could I just use the adverb danas and say ?