To what are people referring when they talk about the Shekhina? Is it a form of God? A part of God? An attribute of God? Perhaps it is something else entirely?
Also, what is the etymology of the word, and is it related to what the Shekhina actually is?
On a simple level, Shechina refers to the manifestation of G-dliness that can be perceived even by created beings. To quote a letter from the Lubavitcher Rebbe:
Nevertheless, G-d desired that the divine influences upon creation... and the divine immanence in the world should also include elements that the human mind can comprehend. […]
This aspect of the divine reality-that which pervades our world to the extent that it can be discovered by G-d’s creatures-is what is meant by the term shechinah, i.e., that which dwells within and enclothes itself.
Etymologically, Shechina comes from the verb "shochen" ("dwells"). It is thus named because it "dwells" in and is "enclothed" in the physical world. Like the verse "v'shochanti b'soicham" - "and I will in them." (See Maamar quoted below.) So yes, its etymology is related to what Shechina actually is.
Kabbalistically speaking, Shechina is, first and foremost, a job description. It refers to when a particular level of G-dliness is "shochen" (or "dwells") in the level below it.
There are several different places where the name Shechina is applied (Maamar Basis Legani 5711 from the Lubavitcher Rebbe):
The initial revelation of Ein Sof, before Tzimtzum (Technically speaking, it refers to the level of Hispashtus HaOr that is the major of Memale Kol Almin)
Kav and Chut in relation to Ein Sof
Malchus of Atzilus in relation to Atzilus (But only when it descends into Atik of Beriah)
The Rebbe comments that it is this last level that Chazal usually refers to as Shechina.
