Rashi (here and here) says it was a fine woolen garment, noting the ketonet passim of Tamar, and noting that pas occurs in the word karpas.
Rashbam translates similarly -- פסים - מעיל.פסים -
לשון כלי מלת.
כמו (אסתר א ו) כרפס ותכלת.
וכמו (שמואל ב' יג יח) כתונת הפסים, דתמר ואמנון.
על שם צרותיו שנמכר לפוטיפר ולסוחרים ולישמעאלים ולמדינים:
fine woolen Heb. פַּסִים, a term meaning fine woolen garments, like“green wool (כַּרְפַּס) and blue wool” (Esther 1:6), and like the fine woolen coat (כְתֹנֶת פַּסִים) of Tamar and Amnon (II Sam. 13:18). The Midrash Aggadah, however, explains that it was called פַּסִים because of his (Joseph’s) troubles, namely, that he was sold to Potiphar (פּוֹטִפַר), to the merchants (סוֹחֲרִים), to the Ishmaelites (יִשְׁמְעִאלִים), and to the Midianites (מִדְיָנִים). [From Gen. Rabbah 84:8]
Ibn Ezra appears to give two perushim:
כמו: פס ידא בלשון ארמית.
But others have different explanations. Thus,
A many-colored coat (37:3)
Ketonet passim, in the Hebrew. The word passim can be translated as "colorful" (Radak; Septuagint), "embroidered" (Ibn Ezra; Bachya; Nachmanides on Exodus 28:2), "striped" (Ibn Janach; Radak, Sherashim), or "illustrated" (Targum Yonathan). It can also denote a long garment, coming down to the "palms" of the hands (Rashbam; Ibn Ezra; Baaley Tosafoth; Midrash Rabbah), and the feet (Lekach Tov). Alternatively, the word denotes the material out of which the coat was made, which was fine wool (Rashi) or silk (Ibn Janach). Hence, Ketonet passim, may be translated as "a full-sleeved robe," "a coat of many colors," "a coat reaching to his feet," "an ornamented tunic," "a silk robe," or "a fine woolen cloak."
(The Living Torah)
though I do not see how they get Rashbam as connecting to "palms."
Shadal has an interesting take, in this context. He says it means coming down to the hands and the feet. Thus:
This then functions as an explanation of why it was favoritism to give him a ketonet pasim. And it also makes sense in terms of Tamar, and all the virgins from bet David:
gedulah. (Of course, other explanations could also work to show such gedulah.)
But this would mean that in the general case, men (based on Yosef) and women (based on Tamar) did not wear clothing which came to the wrists and ankles. Because it got in the way of work, according to Shadal. Or they just didn't, according to Rashbam; Ibn Ezra; Baaley Tosafoth; Midrash Rabbah, and Lekach Tov.
But how could that be? The Avos, and shevatim, surely kept the Torah, with all the proper tznius restrictions. And yet Yaakov's sons and their twin sisters did not wear clothing up to their wrists and ankles. And those girls not from malchut bet David likewise. What would Rabbi Falk, who takes the midrashic example of the Avot, Imahos, and shevatim as a basis for normative halacha / hashkafa in tznius, do?
Alternatively, look to Radak. Rabbi Falk bases himself on Radak's interpretation of a pasuk to show that an engaged couple should ideally not speak to one another at all. Yet here, he says that Yosef as well as Tamar and the rest of the daughters of David HaMelech wore a coat of many colors. How could girls wear coats of many colors? Does this not attract attention? (He might say that they were multicolored in the specific few colors he permits. I don't know. He says (I don't have the full book that black and blue are generally tzniusdik colors, but I think also likes certain "calm" colors.) But then again, the entire idea of a multicolored coat seems to be that it is bright and eye-catching.
Update: I should clarify (and preempt, before any comments arrive) that many possible terutzim come to mind. E.g. everyone else wore knee-highs; these ketonet passim were not so long as to be considered the "very long skirts" condemned by Rabbi Falk; the daughters of bet David were frum, and so they wore it, but so did other frum girls. And so on and so forth. It is very easy to come up with terutzim. But the question should give us pause, and take stock, at the least, before coming up with the terutz. Our our tnius standards identical (in either direction) with those of the Avos?
Update 2: Anonymous (same one? different one throughout? that's why pseudonyms are so great) points out in the comment section the Rabbenu Bachya who cites a midrash (we do not know which) that it is because it was such a large garment that it covered his palms. Presumably this midrash in question is in Breishis Rabba 84:8, which reads (relevant portion in red, and large):
Rabbenu Bachya writes (click here, page 76, midway down the right side -- the text in the image is underlined in red):ועשה לו כתונת פסים
ריש לקיש בשם רבי אלעזר בן עזריה אמר:
צריך אדם שלא לשנות בן מבניו, שעל ידי כתונת פסים, שעשה אבינו יעקב ליוסף, וישנאו אותו וגו'.
פסים שהיתה מגעת עד פס ידו.
שהיתה דקה וקלה ביותר, ונטמנת בפס יד.
שהפיסו עליה, איזה מהם יוליכה לאביו, ועלת ליהודה.
על שם צרות שהגיעוהו,
רבי שמעון בן לקיש בשם רבי אלעזר בן עזריה:
(תהלים סו) לכו וראו מפעלות אלהים.
וכתיב: בתריה (שם) הפך ים ליבשה.
למה וישנאו אותו?
בשביל שיקרע הים לפניהם.
ולשון פסים אמרו במדרש על שם שהיתה גדולה ומכסה שתי פסותיו
In my critical edition of Rabbenu Bachya, the printed text is slightly different, and reads:
ולשון פסים אמרו במדרש על שם שהיתה גדולה ומכסה שתי פסות ידיו
I don't know if that is the meaning of מגעת עד פס ידו. Perhaps he is taking it as ad vead bichlal. Yachol Lihyot, though I would certainly not read it that way. But it seems to me we can read this quite readily into Daat Zekeinim miBaalei haTosafot, who says that the ketonet pasim were סביבות פס ידו. So point taken.
But this is not necessarily what Ibn Ezra means in his second peshat. Or what Shadal means. I agree it would make it difficult to do work, or pretty much anything, so one could well argue that this is what Shadal means. But I am not convinced that this was Shadal's intent. Indeed, it would probably be fairly annoying and not a maaleh. How are you supposed to pick up anything? Write? Turn doorknobs? Long sleeves, especially when they do not fit tightly, can get in the way of serious work plenty. And who says ad means ad bichlal?
Also, if we take it like this, then Rabbi Falk is still in trouble, because if we assume like Shadal and Lekach Tov, then it also was so long it covered the feet (pas regel). And Rabbi Falk writes at length about how the "very long skirt" is untzniusdik, to the extent that in discussing the modesty of Shaul (I have a post about this in the works) in wearing a long cloak, he takes pains to say that it was long, but not excessively long, for that would be untznius.
Regardless of the correct meaning, what is Daat Zekeinim seeing, and what is Rabbenu Bachya seeing, that make them explain like this? One possibility is the gemara in Menachot 11a:
אמר רב חופה שלש אצבעותיו עד שמגיע על פס ידו וקומץ
where it refers to bringing the fingers over against the palm of the hand to do kemitza.
Compare with שהיתה מגעת עד פס ידו. And perhaps they even hand a slight different in nusach in the midrash, replacing עד with על.
On the other hand, let us take a look at Rambam. In Mishneh Torah, when discussing the begadim of the kohanim, he writes:
Would we say that the length of the sleeves of his begadim were such that they covered his palms? This would make the avodah a little difficult, I would think, besides making it look like the kohanim all had bad tailors. And compare with the length of the beged going down -- it reaches to above the ankle. Indeed, at the Temple Institute, they have a picture of the kohanim's begadim (see image to the right of the kohen gadol's begadim) -- and they are even selling them to be stored in closets of kohen hedyots in preparation for mashiach, and they have a picture there, which I reproduce to the right. Note how it does not cover the palms. Does anyone know the literature on this? Do people say the kohen gadol and kohen hedyot had their palms covered?