Parshat Chayei Sarah is a transition parsha. Initially, Avraham is the center of attention. He purchases Ma'arat HaMachpela to bury Sarah. He sends his trusted servant Eliezer to find a wife for his son Yitzchak. For most of the parsha, Yitzchak is in the background. By the end of the Parsha the situation is reversed. Yitzchak marries Rivka, bringing her the tent of his mother Sarah. The emphasis shifts to Yitzchak and Rivka, and Avraham fades from the scene. His death is recorded now, and his burial by his sons Yitzchak and Yishmael, even though Avraham actually lived until his grandchildren Yaakov and Eisav were fifteen.
At the end of our parsha, though (פרק כ"ה), the Torah reveals some new information about Avraham: he marries again to a woman named Keturah, and they have children. (Although Rashi quotes a Midrash to the effect that Keturah was actually Hagar, and that the only women in Avraham's life were Sarah and Hagar, many meforshim disagree, and take the text literally. For example, see רמב"ן, אבן עזרא, רלב"ג, אברבנאל.) Avraham sends his other sons away with gifts while he is still alive, and gives all he has to Yitzchak (i.e., he makes Yitzchak his heir.)
Avraham's motivations for remarrying are discussed by Abarbanel (and somewhat less explicitly by Ralbag.) Avraham was promised by Hashem that Yishmael would do teshuva -- (see Rashi to בראשית טו:טו), but Avraham was still concerned about possible grudges or resentments that Yishmael might harbor against Yitzchak. And so he married Keturah, as a wife (ויקח אשה) and not as a lower status פלגש. Together they raised six sons, and then Avraham sent his sons away.
Now Yishmael could not claim that he had been denied his inheritance. After all, he was the son of Hagar, Sarah's maid, while his six stepbrothers were sons of Keturah, a full-status wife, and they didn't inherit anything from Avraham either.
Avraham was also worried that Yishmael would blame Yitzchak for his expulsion, instigated by Yitzchak's mother Sarah, from the household of Avraham. He therefore sent all his younger sons away as well, with the same kinds of gifts he must have given to Yishmael (אברבנאל). Now Yishmael could understand that it was Hashem, and not Sarah, who directed his being sent away, and that all the other brothers were being treated equally.
The Torah bears witness to the success of this strategy. Yitzchak and Yishmael were both involved in the burial of Avraham next to Yitzchak's mother Sarah in Ma'arat HaMachpelah. Not only that, but Yishmael, the older son, deferred to his younger brother Yitzchak, going against conventional protocol (see בראשית רבה לח,יב).
See the effort expended by Avraham Avinu to make peace between brothers!