What is the difference between "Shevet" and "Mechokek"?
Rashi (from Sanhedrin 5a): This refers to the post sovereign period; 1 "Sheivet" therefore refers to the exilarchs (the heads of Galus in Bavel), 2 and "Mechokek" to the princes in Eretz Yisrael (the head of the Chachamim), who will be always be from Yehudah.
Rashbam: "Sheivet" refers to Yehudah himself, whose brothers appointed him as their king; "Mechokek", to his descendents, who will rule up until Shiloh. 5
Seforno: During the period that he merited Malchus, it will not move from him to another tribe. 6 And from his descendents there will also be Shoftim, at whose feet there will sit the Sofrim of the Dayanim. 7
Targum Onkelos and Targum Yonasan: "Sheivet" refers to kings and rulers; "Mechokek", to teachers of Torah.
Having already discussed the period of David ha'Melech, incorporating the kings of Yehudah.
From the time that David is chosen (See Ramban, as to why Hashem initially picked Shaul - who was not from the tribe of Yehudah).
See Divrei ha'Yamim 1, 28:4. Refer to 49:10:2:2. See Ramban, who explains that after David (with the exception of Yeravam, who was appointed by the Navi Achiyah ha'Shiloni), whenever they crowned a king from any other tribe, they transgressed the express command of Ya'akov Avinu. Indeed, he explains, that is one of the reasons why the family of the Chashmona'im was wiped out.
See also Ba'al ha'Turim.
What is the meaning of the latter half of the Pasuk? What is the significance of Shiloh?
Rashi #1 and Targum Onkelos: Shiloh 1 is synonymous for Mashi'ach. What the Pasuk therefore means is that Yehudah will always reign (in one form of leadership or another) 2 up until the time of Mashi'ach, who will gather all the nations. 3
Ramban: 'The staff of oppression will not cease to function until Mashi'ach comes and breaks the nations, weakening them 4 to the point of the sword, and rendering them unable to fight any further.
Targum Yonasan: 'until the days of King Mashi'ach, his youngest son (David), under whose reign the nations will be crushed.
Rashi #2 (in Divrei ha'Yamim 2, 10:15), Rashbam, Da'as Zekenim #1: Yehudah's descendents will rule over all twelve tribes until the advent of Rechav'am, Shlomoh's son, who will consolidate his kingship in Shiloh. 5 That is when he will lose ten of the tribes to Yeravam. 6
To whom all nations will listen (Targum Onkelos). See Ramban's objection to this translation.
See Ramban DH 'v'Chein Matzasi'.
Shiloh is actually next to Shechem, where they assembled to crown him, and the Pasuk mentions it in honor of the Mishkan, which was there in its time (See Rashbam). 6
See also Ba'al ha'Turim. He will still rule over Yehudah and Binyamin, and part of Shimon and Levi amidst them. (PF)
He explains "Yasur" to mean sprout, and "Yavo" is destruction. Shmuel and Sha'ul reigned after the fall of Shilo and before David, but only for 13 years. (PF)