In the play's prologue, the god Apollo comes out from Admetus' palace in Pherae (modern Velestino in Magnesia), dressed in white and carrying his golden bow, with the intention of leaving to avoid becoming stained by the imminent death of Alcestis, who is being comforted within. He offers an exposition of the events leading up to this moment. He hails the arrival of Thanatos (Death), who, dressed in black and carrying a sword, has come to the palace in his role as psychopomp to lead Alcestis to the underworld. Thanatos challenges Apollo's apparent defense of Alcestis and accuses him of "twisting slippery tricks" when he helped Admetus cheat death in the first place. Apollo reassures him and, in a passage of swift stichomythic banter, proposes a postponement of Alcestis' death, which is sarcastically rebuffed. "For once," Thanatos concludes, "you may not have what is not yours." Defeated, Apollo leaves angrily, prophesying the arrival of a man (Heracles) who will wrestle Alcestis away from Death. Alone with the audience, Thanatos warns that "this was a god of many words; but words / are not enough," before he summons the doors open with the tip of his sword and slowly enters the palace.
The entry of the chorus, or the "parodos" sequence, follows: a chorus of fifteen men of Pherae, led by a "coryphaeus" (chorus-leader), enter the orchestra of the theatre. The chorus-leader complains that they are in a state of suspense, ignorant of whether they ought to be performing mourning rituals for their queen. The chorus' lyrical ode, to which they dance as they sing, consists of two paired stanzas of strophe and antistrophe. They sing of the silence that greets their search for signs of mourning, the evidence of Alcestis' death. "When goodness dies," they lament, "all good men suffer, too." The chorus-leader concludes by dismissing the chorus' search for hope in the situation: "The King has exhausted every ritual."
The first episode begins with a maidservant, who enters from the palace in tears. When the chorus-leader presses her for news, she gives a confusing response: "She is alive. And dead." Alcestis stands, she explains, at this moment on the brink of life and death. The chorus-leader anxiously confirms that all of the customary preparations have been made for her proper burial. The maidservant joins the chorus-leader in praising Alcestis' virtue. She narrates a long description of Alcestis' prayers and preparations to die earlier that morning, when Alcestis cried over the bridal bed that will destroy her, embraced her sobbing children, and bade all farewell. She describes how Admetus held Alcestis weeping in his arms while her eyes clung to the sight of the last rays of sun she would see. The maidservant welcomes the chorus-leader to the palace and goes inside to inform Admetus of their arrival.
Alcestis, on her death-bed, requests that in return for her sacrifice, Admetus never again marry, nor forget her or place a resentful stepmother in charge of their children. Admetus agrees to this and also promises to lead a life of solemnity in her honour, abstaining from the merrymaking that was an integral part of his household. Alcestis then dies.
Just afterwards, Admetus' old friend Heracles arrives at the palace, having no idea of the sorrow that has befallen the place. Unwilling to turn a guest away, the king decides not to burden Heracles with the sad news and instructs the servants to make him welcome and to keep their mouths shut. By doing this, Admetus breaks his promise to Alcestis to abstain from merrymaking during the period that follows her death. Heracles gets drunk and begins to irritate the servants, who loved their queen and are bitter at not being allowed to mourn her properly. Finally, one of the servants snaps at the guest and tells him what has happened.
Heracles is deeply embarrassed at his blunder and his bad behaviour and he decides to ambush and confront Death when the funerary sacrifices are made at Alcestis' tomb. When he returns, he brings with him a veiled woman whom he tells Admetus he has won in a competition. He asks his host to take her and look after her while Heracles is away on his labours. After much discussion, he finally forces a reluctant Admetus to take her by the hand, but when he lifts the veil, he finds that it appears to be Alcestis, back from the dead. Heracles has battled Death and forced him to give her up. She cannot speak for three days, after which she will be purified and fully restored to life.

Question: What did Alcestis make her husband promise at her death bed?
Answer: never marry again, don't forget her, don't let stepmother be in charge of her children
Alternative answer: Never to remarry, nor forget about her, nor place a stepmother in charge of their children.