It means that the university courses will start in three weeks.

To answer the question, I will reverse the direction of the translation :

• The first sentence :

  The university courses will start in three weeks

  becomes

  Les cours à l’université vont commencer dans trois semaines

  so, within 21 days.

• The second sentence :

  The university courses will start in week three

  becomes

  les cours à l’université vont commencer à la troisième semaine

  That’s “correct” but case-specific : indeed there is a lack of information here, since we do not know which month or year it is about (third week of which month)… Thus the speaker has to precise a landmark starting from which we start counting weeks.

A more convenient sentence would be :

Les cours à l’université vont commencer à la troisieme semaine du mois prochain.

I hope this clears up the ambiguity.

Finally, in French or in English it’s not convenient (although correct) to reference order by number such as in “week three” : we should rather use “third week”/”troisième semaine”.