Learn Russian: Ещё один день.

The word ещё usually means "still" or "yet," but when you put it right before a number, it means "more" or "additional." So ещё один translates to "one more."

Why is the word for "one" один and not одна?

The word for "one" must match the gender of the noun it counts. Because день (day) is a masculine noun, we use the masculine form один. If you wanted to say "one more week," you would use the feminine form: ещё одна неделя.

Could this also translate to "another day"?

Yes, but only when you mean "one additional day." If you mean "a different day" (for example, "Let's do this another day instead"), you would use the word другой instead.

Is this a complete sentence without a verb?

Yes! In Russian, short statements of fact or thoughts often do not need a verb. The existence of the day is simply being stated.