Learn Swedish: Lärarens dator är ny. - The teacher's computer is new.

Because it combines "the" and the possessive -s. The basic word is lärare (teacher). To say "the teacher", you add an -n to get läraren. Finally, you add the -s to make it "the teacher's" (lärarens).

If it is "the" teacher's computer, why isn't it datorn?

In Swedish, just like in English, when a noun is owned by someone using the -s, it stays in its basic, indefinite form. You say lärarens dator (the teacher's computer), never lärarens datorn. The ownership word already makes it specific enough!

What would the sentence be if it was "A teacher's computer is new"?

You would drop the definite -n from the owner. "A teacher" is en lärare, so "a teacher's" simply becomes en lärares. The whole sentence would be En lärares dator är ny.

Why do we use ny here instead of nytt or nya?

Adjectives have to match the noun they describe. Because dator is an en-word (en dator), the adjective takes the basic form ny. If we were talking about an ett-word like ett hus (a house), we would use nytt (Lärarens hus är nytt). If it were plural (computers), it would be nya.