Learn Icelandic: Ég þekki kennarann mjög vel.

Kennarinn means "the teacher" in the nominative (subject) case. But here, the teacher is the direct object receiving the action of the verb þekki (know), so it must be in the accusative case. For the masculine noun kennari, the accusative form is kennara, and adding the definite suffix ("the") makes it kennarann.

I've seen veit used for "I know". Why is it þekki here?

Icelandic uses different verbs for "to know". You use þekkja (here as ég þekki) when you are familiar with a person, place, or thing. You use vita (like ég veit) when you know a fact or a piece of information. Since you are talking about knowing a person, þekki is the correct verb.

Does the verb þekkja always make the noun after it accusative?

Yes! The person or thing that you are familiar with is the direct object of the sentence, so it will always take the accusative case.

Could I say Ég þekki mjög vel kennarann instead?

While Icelanders might understand you, it sounds unnatural. In Icelandic, adverbs describing how you do something (like mjög vel) usually go at the end of the clause, after the direct object (kennarann).