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.
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).
Hadar Zekeinim: Maharam says that this is the ability to impose oaths on Shedim to obtain whatever they wanted. R. Yakov of Motrovil says that he taught them to give disgraceful names to their idolatry, which they used to call "Hashem". (Why is this called 'gifts'? - PF)
from which he did not want to benefit - based no doubt, on the Pasuk "v'Sonei Matanos Yichyeh" (Mishlei 15:27). Kli Yakar - what he did not work for is not called "Asher Lo."
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 (in Aram Naharayim) 1 from his family.
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.)
רש"י: שלא היתה אלא פלגש אחת: אם הייתה רק אחת למה כתוב בלשון רבים?
גור אריה: פירוש הכתוב 'בני פילגש אחת מן הפילגשים דעלמא'. 1
כלומר- זה מתייחס לכלל הפילגשים ולא אל פילגשו של אברהם במסוים.
רש"י: נשים בכתובה, פילגשים בלא כתובה: רמב"ן: כתובה היא תקנת חכמים, ופילגש היא בלי קידושין?
גור אריה: קשה לומר כרש"י שבלי כתובה היא נקראת פילגש, שהרי ממון ניתן למחילה והוא קידש אותה על מנת שאין לה עליו כתובה. 1
ואין חסרון באישות שלהם אם הוא קידש בלי כתובה.
רש"י: שם טומאה מסר להם: הכוונה למעשה שדים וכשפים כפי שפירש רש"י בסנהדרין (דף צא.) וקשה שאסור להשתמש בכשפים?
גור אריה: מסר להם שם טומאה על מנת לסלק מהם כשפים, ובשדים אפשר שמותר להשתמש.
רש"י: נתן אברהם מתנות - פירשו רבותינו וכו': למה לא פירש "מתנות" כפשוטו?
גור אריה: אברהם כבר נתן את כל נכסיו ליצחק.