"nap" is day, napunk is "our day". If you want to learn more about how possession works in Hungarian take a look at this link: http://www.hungarianreference.com/Possession/SingleObjects.aspx
Hungarian doesn't have the verb "to have", but express it with "van" + object with some pronoun ending. For example:
- Van macskám = "I have a cat" or, literally, "There's a cat-my"
There is "van" and the object ("macska") with an ending (-m is for "I"; -d is for "you", etc.) and a linking vowel that can change depending on the noun. So, in the sentence above:
- Szép napunk van ma = maybe "there's a beautiful day-us today"
There's the "van" and "nap" with the personal ending for "we" (-nk/-unk/-ünk…)
I think that "Today is a beautiful day" translates to "Ma (egy) szép nap" but I'm not sure.
I find the "-unk" construction to be convenient to think of as "of ours". (This is one completely legitimate translation of it.) So "szép napunk van ma" would translate through this lens as "It is a nice day of ours today."
Similarly with the "-om" and other forms of the possessive.