Isaac is born and names in verse 3 of Chapter 21, and if you recall the earlier studies the reason he is named "Isaac" was due to the incredulous response of Sarah when she was told she would give birth to him - she laughed hysterically, as it was unthinkable at her age to give birth to a baby(note verse 6)! But, God did as He promised, and Isaac was conceived, and also delivered as a healthy, normal baby boy. Abraham, the proud dad, was 100 at the time of Isaac's birth, and Sarah would have been well into her 90's. But, in these early days of human history, longevity and things like this were more common due to the fact that lifespans were not totally affected yet by the consequences of the Fall, although later they would be drastically shortened. As people were living longer, it is of no real mystery that they were probably healthier too, which is why Sarah was able to carry a baby to term and give birth at such an advanced age. In verse 4, Abraham as part of an earlier covenant has Isaac circumcised when he is eight days old, a ceremony that many religious Jews carry on today which is called a bris.
There is a fast-forward on the story to maybe a couple of years later in verse 8 - Isaac is now a toddler, and Sarah has weaned him from nursing, which of course means a proud dad like Abraham has to throw a party - this is the promised son, so life events are occasions for celebration. At some point around the same time one day, Ishmael (who is called in this passage "the son of Hagar the Egyptian") is observed by Sarah "playing" with Isaac (v. 9) and she becomes very upset by it. Now, as we read this, we would ask the question as to why Sarah would be so upset with the boys playing normally? However, a little more research revealed some theories on the source of her concern. One thing that is mentioned by an Orthodox Jewish source I looked at (www.Chabad.org - accessed 8/31/2016) suggests that Ishmael, because of his own Egyptian mother Hagar's inclinations, was inclined to idolatry and that Sarah didn't want Ishmael to corrupt Isaac and jeopardize the Covenant. Other Bible translations also reveal something as well. For instance, the NIV translates the word "playing" as "mocking," and David Stern's Complete Jewish Bible renders it as "making fun of." The ancient text of the Book of Jasher, in 21:13-15, reveals something even more serious - it suggests that Ishmael was trying to actually kill Isaac with a bow as a cruel sport (Ken Johnson, Ancient Book of Jasher - Biblefacts Annotated Edition. Lexington, KY: Biblefacts, 2008. p. 44). Likewise, the later Book of Jubilees 17:4 notes that Sarah's concern was based on the fact that Abraham still cared a lot for Ishmael (Ishmael was his son too, after all!) and she was jealous because she feared Isaac would lose out (George H. Schodde, trans. The Book of Jubilees - From the Ethiopic. Oberlin, OH: E.J. Goodrich, 1888. p. 45). It is my theory personally that all of this simultaneously was true at the same time, and that each source and translation just took the same situation from a different angle. Therefore, the synthesis I would propose based on these observations is this - up until this point Ishmael was the older son, and perhaps in Abraham's mind (despite the fact he knew better) he was grooming Ishmael as his heir, but Sarah saw God work and wanted to protect Isaac. Although perhaps Ishmael was innocently playing with the infant Isaac at this point, it is also possible he may have gotten a little rough in his play with the baby, and that sort of stirred Sarah up too. Whatever the case, this was the motivation for what happened next in the story.