Learn Croatian: Vlasnik stana je ovdje. - The owner of the apartment is here.

Because it is in the genitive case, which is used to express possession or the English word of. The dictionary form is stan, which is a masculine noun. To make it genitive, we add -a to the end, making it stana.

Can I change the word order to stana vlasnik?

No, it sounds very unnatural. In Croatian, the genitive noun almost always comes right after the noun it describes. Just like in English, we say the owner of the apartment, so vlasnik comes before stana.

Why doesn't the verb je come right after Vlasnik, since short words usually want the second position?

The short verb je (is) does want to be in the second position. However, Croatian grammar treats the phrase Vlasnik stana (the owner of the apartment) as a single, connected block of information. Therefore, je counts that whole phrase as the first position and places itself right after it.

I have seen tu used for here. What is the difference between tu and ovdje?

Both mean here, but ovdje specifically means right here, exactly where the speaker is. Tu technically refers to a location closer to the listener, though in everyday casual speech, people often use them interchangeably.