Who are the Pilagshim (concubines) mentioned here?
Rashbam, Ramban and Targum Yonasan: Hagar and Keturah.
Since "ha'Pilagshim" is written without a 'Yud' (before the 'Mem', to denote singular), though our texts do have one.
Why does it refer to Keturah, whom Avraham married (verse 1), as a Pilegesh?
Ramban: Since the Torah wrote "because in Yitzchak (exclusively will be called offspring", any other wives that Avraham had were automatically referred to as Pilagshim.
Ha'Emek Davar: The custom was that a man drew close his children from his esteemed wife, and distanced Bnei ha'Pilagshim and they would not inherit (like Yiftach would not), even though this is unlike Torah law. Therefore Bnei Keturah are called Bnei Pilagshim, even though she was not a Pilegesh, just like Bnei ha'Nevi'im were not sons of Nevi'im.
Bearing in mind the previous Pasuk, which says that he gave everything to Yitzchak, what sort of 'gifts' did Avraham give the sons of the Pilagshim?
Rashi #1: The name of Tum'ah. 1
Rashi #2: All the gifts that he received on behalf of Sarah as well as all the other gifts that were given to him. 2 Everything that he earned by the toil of his hands, he gave to Yitzchak.
Targum Yonasan: He gave them property and moveable goods - now, in his lifetime.
Rashbam: He gave them much money (in addition to what they would inherit in Avraham's place of origin - refer to 25:6:6:1).
It says "Nasan" (he had 3 given) to Bnei ha'Pilagshim, i.e. before he sent them away. Va'Yiten (he gave, i.e. now) all that he had [left] to Yitzchak. (PF)
Michtav me'Eliyahu (5, p.462, from the Zohar): He gave to them Chachmah of this world. This was Chachmas Bnei Kedem (Melachim 1:5:10).
What is the definition of a 'Pilegesh'?
Rashi: A woman who is married (with Kidushin), but who lives with her husband without a Kesubah. 1
Ramban: A woman who lives with a man without Kidushin.
See Ramban, who disputes Rashi's version of the Gemara in Sanhedrin (Daf 21a).
Why does the Torah add the words "whilst he was still alive"?
Seforno: Because he did not want to rely on any of the various ways of bequeathing after his death.
Why did Avraham send the Bnei Keturah to 'the land of the east'?
Rashbam: To take over the lands that he inherited there 1 from his family.
Chizkuni, Malbim, Ha'Emek Davar #1: It was lest they argue with Yitzchak about inheritance of Eretz Yisrael. 2
Ha'Emek Davar #2: He sent them east of the east land, to distance them from worshippers of the sun and nature. In Kedem, they used witchcraft to overcome nature. Bnei Noach are not commanded about witchcraft. Even the opinion that disagrees, he agrees that it is not at severe as idolatry.
The Torah does not mention Bnei ha'Pilagshim again. What happened to them?
Michtav me'Eliyahu (5, p.461): They separated from Kedushah and joined to the other nations. However, Avraham cast many sparks of Kedushah into Yishmael and Bnei Keturah, and Yitzchak into Esav. Yisrael needed them - to afflict us when needed, or that we should learn from their afflictions - "Hichrati Goyim... Tikchi Musar" (Tzefanyah 3:6,7).
Rashi said that "ha'Pilagshim" is written without a 'Yud', to teach that there was only one. If so, there should be no Mem!
Peninim mi'Shulchan ha'Gra: A man may not keep a wife without a Kesuvah. Why is it not called a Kesav? There is a Yud in Ish, and a Hei in Ishah. Vov and Hei were put into the word Kesuvah, to complete Hashem's name. Rashi said that a Pilegesh has no Kesuvah. Without the Yud, we read Palag Shem, i.e. only half of Hashem's name is there. (In a true marriage with Kesuvah, His full name is there.)
QUESTIONS ON RASHI
Rashi writes that Avraham gave Shem Tum'ah to Bnei ha'Pilagshim. What is this?
Chizkuni #1: It is a name of Hashem that they can say even when Tamei, in order to impose oaths on Shedim. 1
Chizkuni #2: It is a name (not of Hashem) that they can say to impose oaths on Shedim.
Malbim: He knew that they will be under Mazal and stars, and gave them names to rule over them. This is not proper to keep, rather, to give to others.
Chizkuni: Some reject this. Heaven forbid, Avraham would not give a Kadosh name to Resha'im! (What is the source that they were Resha'im?) (PF)