Sure enough, Sarah gets pregnant and gives birth to a son.
That's right—the 90-year-old lady had a baby. The narrator emphasizes that this is in accordance with what the deity has been saying all along (remember 15:5; 17:16-19; 18:10?)
Abraham names his son Isaac, which in Hebrew means "one who laughs." He did make his parents laugh quite a bit before he was born.
Abraham circumcises Isaac when he's eight days old—as per his agreement with God—and he celebrates Isaac's weaning with a party.
Sarah sees Ishmael, who is not named here, but called, "the son of Hagar, the Egyptian, whom she had borne to Abraham" (21:9 NRSV). These epithets in the place of the name help us get why Sarah is so upset.
Ishmael is "playing" (21:9 NRSV) or "mocking" (KJV) or simply "laughing." The real point is that the verb used is formed from the same root as Isaac's name and means, "to laugh."
Sarah tells Abraham to eject "this slave girl with her son" (21:10 NRSV). Still no proper names being used.
God tells Abraham to listen to Sarah and not to worry.