I have frequently heard the human (or only the Jewish?) soul referred to as "חלק אלוק ממעל", literally "portion/section/piece/allotment of God (from) above".
The first source for that expression is the Alshich - first published 1593, but likely said significantly earlier. The concept is older. According to Tanya it means that the soul uniquely feels itself inseparable from G-d, more so than other things that exist in creation.
As far as I know, the first source for this specific expression of the idea is in the Alshich (first published in 1593, but based on drashos he would have given over time), in a few places. One reference is here. The Kli Yakar (published 1602) also has several mentions of it, one is Bereishis 1:31 and another is Bereishis 9:20.
As far as I know the first appearance of the famous addition of the word "mamash" that is known from Tanya at the beginning of chapter two is in a sefer called שפע טל, by Rabbi Shabtai Shefel Horowitz published in 1612. In the first paragraph of the introduction he writes:
ידוע שהנשמות של אומה ישראלית הם חלק אלוה ממעל על זה רמז הפסוק כי חלק ה' עמו ר"ל חלק ממש כחלק שנחלק מאיזה דבר הוא שוה ודומה בעינו לאותו דבר שנחלק ממנו
It is known that the souls of the Jewish nation are a "portion of G-d above." On this the Posuk hints "A portion of G-d is with him" meaning an actual part [as opposed to an idiomatic statement] that was taken from something - it is equal and similar in its type to the thing it was taken from.
It is certainly understood from the language "it is known" that the concept is not new even at that point, even if the association with this particular wording is. (See, for example, the Ramban on Bereishis 2:7 where it says that the soul is רוח שם הגדול - which is in line with how Tanya explains it).
The author of the שפע טל is the nephew of Rabbi Yishaya Horowitz, better known as the Shaloh HaKadosh. In his Sefer (Part 1, Bais Chachma towards the end) he says "You already know the Neshama is a Chelek Elokah Mimal". This was printed later than the שפע טל (well after the Shaloh HaKadosh had passed away).
In terms of what it means, this is explained at some length in Tanya Part 1 Chapter 2 as well as Part 3 Chapter 4. That latter explanation is more direct and to the point: The soul is G-dly, a part of G-d's Name. It is a metaphor that expresses the idea that there is a qualitative difference between the soul and other spiritual creations like angels. Lessons in Tanya succinctly sums it up:
This, then, is the difference between souls and angels: Souls derive from the innermost aspect of G‑dliness, the Tetragrammaton, while angels are rooted in the external aspect of G‑dliness, the Divine Name Elokim
What characteristics of the soul also characterize Havayah, the Four-Letter Name of G‑d, and thus indicate that the soul is indeed a part of that Name? In answer to this question, the Alter Rebbe now explains that just as the Ten Sefirot are included within the Tetragrammaton, so, too, there are ten corresponding faculties that are intrinsic to the soul.
So to sum it up in a sentence, although all creation comes from G-d, creation in general feels itself to have an independent existence, whereas the soul spoken about here does not - it recognizes its inseparability from G-d.
In Tanya the Alter Rebbe notes an observable expression of this phenomena - many Jews throughout history who were emphatically not observant to any serious degree, and simply didn't do much about their Judaism in their day to day lives still gave up their lives rather than undergo forced conversion. This is an expression of the soul's refusal to be separate from G-d above its own existence. [This is sometimes misunderstood to mean that martyrdom is uniquely Jewish. It is not at all, but the point being a martyr is someone you would expect to have a high dedication to their cause prior to the circumstance where they chose martyrdom, rather than it coming seemingly out of nowhere].
Basically, as I heard Rabbi Elimelech Zweibel explain it, the problem with the metaphor of the שפע טל above is that when it comes to something physical, you can say that part is taken, and moved to another place. Take a cup of water out of the ocean, and bring it to your house. You now have a piece of the ocean in your house.
However, in these spiritual concepts of higher angels, lower angels, etc. they are not physical. You cannot "move" them to a different place. When something is of a lower spiritual stature, it is in the lower world by definition. If you take something spiritually higher and lower its spiritual status, it is inherently lower. So how can anything spiritual retain that higher world aspect and still come down into lower worlds.
For this the Alter Rebbe gives two metaphors. One is speech vs. blowing, the other is fathering a child. The point of both is to illustrate something descending into a lower state while maintaining qualities of its higher source, different from other things that come from the same source but do not maintain that quality.
its a quotation from Scripture (Iyov 31:2)
and see tanya, (by rabbi shneur zalman boruchovitch - first lubavitch rebbe ) those are the first words of chapter two: (with the addition of the word "mamesh" ) it is explained there a length.
here is a quote
to describe G-d’s implanting the Jew’s soul in his body signifies that this soul originates in the “innermost” aspect of G-dliness.
see there at length
The Ramchal in Da'at Tevunot refers to the soul as a chelek eloah mima'al. The reference is in siman 24 (page 7) of the R' Chaim Friedlander edition. I don't know if there are any sources older than that, but it sounds from the Ramchal's verbiage that the concept was already pretty well known.
Nefesh Hachaim Shaar 1 Chapter 5:
שהאדם השלם כראוי עיקרו הוא נטוע למעלה בשרש נשמתו העליונה ועובר דרך אלפי רבואות עולמות עד שקצהו השני הוא נכנס בגוף האדם למטה. זהו כי חלק ה' עמו יעקב חבל נחלתו שעיקרו קשור ונטוע למעלה חלק הוי"ה ממש כביכול. ומשתלשל כחבל עד בואה לגוף האדם
Man, appropriately perfected, has his roots planted above in the supernal source of his soul, and passes through millions of worlds until it reaches the other extreme and enters the body below ... his main element is connected and planted above, a part of Hashem, as it were (meaning, not literally). And it lowers like a rope until it reaches the body.
This is a brief snippet, see it inside for yourself from the beginning of the chapter for the full picture, but Nefesh HaChaim understands this idea to mean that the soul of man is the epicenter of creation and is the guiding force that affects all of existence. It is a description of the loftiness of the position which the purely spiritual soul of man occupies.
It does not mean that the soul of a person is a literal part of the essence of Hashem - in fact, the Nefesh HaChaim felt this was heresy to imply in any fashion that Hashem exists on a plane to which we can relate, as did the Vilna Gaon in his famous letter detailing his issues with Chassidic philosophy.
R. Yosef Yaavetz (c. 1440-1508) uses this phrase in his commentary on Avos 1:18:
ב' חלקים יש בנו, חלק המלאכים והוא השכל, וחלק אלוה ממעל, כי רוח מלפני יעטוף, והדבור שליח השכל, והמעשה שליח החלק האלהי, ושניהם צריכים לעבודת הש"י, הוא מה שאמר ואהבת את ה' אלהיך בכל לבבך הוא השכל, בכל נפשך המעשה המורה על החלק האלהי
Also prior to Alshich is Maasei Hashem by R. Eliezer Ashkenazi (Bereishis ch. 17), who lived from 1512-1585 (roughly the same as the Alshich). Maasei Hashem was first published in 1583, ten years before the Alshich:
עוד השמיענו נמשל עמוק נמשך מן המעשה הזה, כי האדם נברא בשני יצרים יצר הטוב ויצר הרע, ונברא מן החומר וניתן בו חלק אלוה ממעל היא הנשמה שעליה נאמר (בראשית ב, ז) נשמת חיים
Mateh Moshe, first published in 1591, writes:
כי הנשמה החשובה חצובה מתחת כסא הכבוד, חלק אלוה ממעל, נקרא פמליא שלמעלה, והגוף הנגוף הלז עפר מן האדמה נקרא פמליא שלמטה.
This phrase in reference to the soul also appears in numerous places in the Shach al haTorah, first published in 1605. While not the first source, it is worth noting that the Gra (Aderes Eliyahu, Bereishis 1:2) cites this in the name of רז"ל (our sages of blessed memory):
ורוח אלהים. הסכת ושמע כי כל הנמצאים למעלה מן הרקיע ואשר יגיע לנו מהם נקרבים ונסמכים למלת אלהים מפני שהם עושי דברו. בלתי פונים אל דבר מקביל רצונו. וכן ארון אלהים (ש"א ד') לפי שהוא דעת אלהים והמקיימים אותה עושים רצון אלהים. וכן ואדוני המלך כמלאך אלהים (ש"ב י"ט) כחכמת מלאך אלהים (שם י"ד) וכן אל שמעי' איש האלהים (מלכים א' י"ב) ויאמר אל איש האלהים (שם י"ג) הוא מונח על איש המיחד רעיוניו ומגמתו לעשות רצון אלהים. וכן רוח אלהים הוא הרוח השוכן למעלה מהרקיע אשר עלינו וזה שאמרו רז"ל שנשמת ישראל חלק אלוה ממעל. ביאורו שהוא חלק מה מרוח אלהים השוכן למעלה
The Ohr HaChaim (Bereishis 1) also uses this phrase:
בחינת הנשמות הקדושות הם בחינת אור התורה ובעל נפש הוא בעל תורה, והוא אומרו ויאמר אלהים יהי אור שהיא הנשמה שאורה רב להיותה חצובה מתחת כסא כבודו יתברך חלק אלוה ממעל שולחה ה' להאיר אל האדם לבל יטבע בטיט היצה"ר
Likewise, Ohr HaChaim (Shemos 20:20):
כי כבר הודיענו הכתוב כי נפשות ישראל חלק אלוה ממעל דכתיב (דברים לב ט) חלק ה' עמו
