Learn Hardcore Greek: Σήμερα είναι πολύ όμορφη μέρα.

Greek does have an indefinite article (ένας, μια, ένα), but it is very often left out in sentences with είμαι / είναι when you are simply describing what something is.

So instead of literally saying Σήμερα είναι μια πολύ όμορφη μέρα, Greek can just say Σήμερα είναι πολύ όμορφη μέρα and it still means Today is a very beautiful day.
The structure [subject] + είναι + [adjective + noun] often does not need ένας / μια / ένα when it’s a general description.

If I say Σήμερα είναι μια πολύ όμορφη μέρα, does the meaning or feeling change?

Yes, it is also correct, and many speakers actually say it this way very often.

Σήμερα είναι πολύ όμορφη μέρα. – neutral description: Today is a very beautiful day.
Σήμερα είναι μια πολύ όμορφη μέρα. – almost the same, but μια can make it feel a bit more like what a very beautiful day (this is) or today is one very beautiful day – slightly more vivid or emphatic.

In everyday conversation, both versions sound natural; the difference is small and mostly about nuance, not grammar.

Why does όμορφη end in -η and not in -ος?