I would like to know if the "deep sleep" of Abraham (Genesis 15:12) is literally a "sleep", or a state of "trance" as The Midrash of Philo says (15:12). What does the biblical term mean? Also, according to Daniel 8:18, the prophet fell to the ground. Are we meant to think that Abraham also fell to the ground?
The commentaries highlight that it definitely wasn't the normal type of sleep.
The Da'as Zekeinim writes that he was clearly experiencing a prophecy:
During this sleep G–d showed Avraham that his descendants would endure the enslavement to several kingdoms at different times as well as the collapse of these kingdoms (empires)...
The Bechor Shor explains the mechanics of this sleep:
ותרדמה נפלה על אברם. מאימת השכינה שבאה לעבור בין הבחרים
A deep sleep fell upon Avram: From when the Divine presence (Shechina) comes to pass amongst those who have been chosen.
So this sleep was the means through which he experienced G-dly interaction.
The Malbim goes as far to say that this sleep was so deep that it deprived him of any bodily strength - "ממצוקת לב", due to heart failure - i.e. he was so pained about this distressing prophecy and the bleak picture it painted.
As a good overview, the Midrash in Bereishis Rabbah 17:5 helps provide a good summary as to how we can understand this term "תרדמה" or "deep sleep":
רַב אָמַר שָׁלשׁ תַּרְדֵמוֹת הֵן, תַּרְדֵּמַת שֵׁנָה, וְתַרְדֵּמַת נְבוּאָה, וְתַרְדֵּמַת מַרְמִיטָה. תַּרְדֵּמַת שֵׁנָה וַיַּפֵּל ה' אֱלֹהִים תַּרְדֵּמָה עַל הָאָדָם וַיִּישָׁן. תַּרְדֵּמַת נְבוּאָה (בראשית טו, יב): וַיְהִי הַשֶּׁמֶשׁ לָבוֹא וְתַרְדֵּמָה נָפְלָה עַל אַבְרָם. תַּרְדֵּמַת מַרְמִיטָה (שמואל א כו, יב): אֵין רֹאֶה וְאֵין יוֹדֵעַ וְאֵין מֵקִיץ כִּי כֻלָּם יְשֵׁנִים כִּי תַּרְדֵּמַת ה' נָפְלָה עֲלֵיהֶם. רַבָּנָן אָמְרֵי אַף תַּרְדֵּמָה שֶׁל שְׁטוּת, דִּכְתִיב (ישעיה כט, י): כִּי נָסַךְ עֲלֵיכֶם ה' רוּחַ תַּרְדֵּמָה.
Rav said that there are three types of 'deep sleep' and they are: 1) A 'tardeima' of deep sleep, 2) A 'tardeima' of prophecy, and 3) A 'tardeima' of almost 'dead-like' slumber 1.
*A 'tardeima' of sleep 2 - e.g. (Bereishis 2:21) "So the LORD G-d cast a deep sleep upon the man; and he slept". 'Tardeima' of prophecy - e.g. (Bereishis 15:12) "As the sun was about to set, a deep sleep fell upon Abram" A 'tardeima' of 'dead-like' slumber - e.g. (Shmuel I 26:12) "No one saw or knew or woke up; all remained asleep; a deep sleep from the LORD had fallen upon them.
The Rabbis said there is even (a fourth); a tardeima of nonsense, as it writes, (Yeshaya 29:10) "For the LORD has spread over you A spirit of deep sleep".
So it seems from this overview that this instance of Avraham was one of prophecy.
The one alternative approach, is to understand this incident as a literal deep sleep as he recovered from a period of intense prayer. In Pirkei DeRabbi Eliezer 28:9 it writes:
A deep sleep fell upon him, and he slept, as it is said, "A deep sleep fell upon Abram" (Gen. 15:12). Does then a man lie down and sleep, and yet be able to pray? But this teaches thee that Abraham was lying down and sleeping because of the intensity of his prayer that his children might enslave these four kingdoms...
1 For more discussion on this third category refer to here
2 In the case of Adam, the commentaries (refer to Sforno, Chizkuni and Malbim amongst others) mention that this deep sleep was to serve the purpose a bit similar to a modern anaesthetic so that Hashem could extract the צלע without causing Adam any undue pain or fear.