Theseus is the duke of Athens, a warrior who won Hippolyta, queen of the Amazons, in battle.  Egeus is the father of Hermia, the daughter who wants to marry Lysander, not Demetrius who is her father's choice of husband for her.  Egeus is furious that his daughter is bucking his will and determination that she will marry the man he wants her to marry.  He angrily brings his daughter the two men to Theseus and asks Theseus to impose "...the privilege of ancient Athens.." which is to make Hermia marry the man her father chooses or be put to death for the refusal.  He seems very determined that this be carried out; a rather harsh reaction to his daughter's preference for another husband.  Theseus isn't quite as blood-thirsty as Egeus (having been softened perhaps by his love for and impending marriage to Hippolyta).  He tells Hermia that she has 3 choices: marry Demetrius like her father wants, become a cloistered nun forever shunning men, or die.  When the situation seems to resolve itself in Act 4, sc. 1, and Demetrius now declares his love for Helena rather than Hermia, Theseus seems pleased to celebrate their weddings as opposed to enforcing any punishment or execution.