The Body
The body is the physical part of a person, where the neshamah (soul) resides while in this physical world.
The choice of the Yid (Jew) is specifically from the perspective of the body, since regarding the neshamah the concept of choice doesn't apply, as there is no comparison between the neshamah of a Yid and the neshamah of a non-Jew. Specifically from the perspective of the body, where there is no difference between a Yid and a non-Jew, this is where Hashem's essential choice lies.
Not only that, but when Hakadosh Baruch Hu provides life force to the body of a Yid, He provides it with love and desire - an aspect of "panim" (Divine countenance), in contrast to the bodies of the nations of the world.
Among the sayings of the Alter Rebbe: "Men hot gar kein hasaga nit, vi es iz teier ba Hashem Yisborach der guf fun a Yidden" (There is no comprehension of how precious the body of a Yid is to Hashem).
The Chassidic Approach to Refining the Body edit
In contrast to the Mussar approach which advocates breaking the body through mortification and fasting, Chassidus explains that one should harness the body for the needs of avodas Hashem (Divine service), as the Rambam states "having a healthy and complete body is among the ways of Hashem." As the Baal Shem Tov explains on the verse "When you see the donkey of your enemy collapsed under its burden, and you would refrain from helping him, you shall surely help along with him" - its interpretation is: when you see the materiality of the body - which is your enemy, and you think you need to break it, you should know that you need to help along with it, specifically with the body.
However, Chassidus explains well that the body is "mashcha d'chivya" (the skin of the snake), and its vitality comes from the nefesh habehamis (animal soul) in the body (for a beinoni, unlike tzaddikim, for whom the body is like a stranger, as Hillel would say when going to the bathhouse that he was going to do chesed with "the wretched one"). Moreover, it interferes with avodas Hashem due to its coarseness and materiality. However, the ways of breaking the body in Chabad Chassidus are specifically through spiritual contemplation of human lowliness and the evil of one's deeds, not through physical breaking of the body.
The body's root in Elokus (Divinity) is from the aspect of "the face of man" in the Divine chariot.