According to Genesis 49:10, when Yaakov was at his deathbed, he called in his sons and desired to bless them. At Judah's turn, his father said that the scepter should not depart from Judah, nor a lawgiver from between his feet, until Shiloh should come; and to him should the gathering of the people be. What is the meaning of Shiloh here in this prophecy?
Besides the other answers given here, I would add the following (which I develop at greater length here):
The ketiv of the word is שילה, Sheilah, who was Yehuda's youngest son. The entire pasuk is a coded reference to the incident with Yehuda and Tamar. Recall that he gave Tamar his staff, his signet ring, and his identifying cord, as surety for payment. And that this only came about because he held back his youngest, Sheilah.
לֹא-יָסוּר שֵׁבֶט מִיהוּדָה: the staff shall not depart from Yehuda וּמְחֹקֵק: and his engraved signet ring מִבֵּין רַגְלָיו: from between his legs עַד: until he said כִּי-יָבֹא שִׁילֹה: that Sheilah would eventually come.
Note that other parts of Yaakov's blessing are also coded references to incidents of the shevatim. For instance, גּוּר אַרְיֵה יְהוּדָה, מִטֶּרֶף בְּנִי עָלִיתָ as a reference to the sale of Yosef; and כִּי עָלִיתָ מִשְׁכְּבֵי אָבִיךָ; אָז חִלַּלְתָּ, יְצוּעִי עָלָה as Reuven and Bilhah; and כִּי בְאַפָּם הָרְגוּ אִישׁ, וּבִרְצֹנָם עִקְּרוּ-שׁוֹר as a reference to Shimon and Levi wiping out Shechem.
As for the contentions with Christian interpretations of the verse, see here.
A sampling of other explanations
Then there are the various detailed explanations that do refer שילה to Moshiach (as WAF noted), on various grounds. (According to all of them, the prophecy means not that Moshiach's kingship will mark the end of Yehudah's primacy, but on the contrary - that this will be its climax.)
Well, since I haven't had the responses I expected, here is my contribution in terms of what I think about the prophecy:
Shiloh - Genesis 49:10
When Jacob thought he was about to die, he invited all his sons unto himself at his deathbed for the last blessings. At the turn of Judah, Jacob said that the scepter would not depart from his Tribe until Shiloh came.
Christians in general assume that's a prophecy about Jesus, and I have researched about the matter, and happened to have found out that's not true.
The Tribe of Judah had grown to become the leader over all the other Tribes, and kept the monopoly to exert hegemony over them all. That's the scepter that would not depart from Judah till Shiloh came.
After the death of Solomon, the Prophet Ahijah from Shiloh took his coat and went out to meet Jeroboam, who was the leader of forced labor among the Northern Tribes. As the Prophet met Jeroboam, he tore his coat in twelve parts and gave ten to Jeroboam, saying that God had decided to split the Tribes in two Kingdoms, and that ten of those Tribes would be governed by Jeroboam. That's when Shiloh came, and Judah lost the hegemony over ten of the Tribes. (I Kings 11:29-32) It's important to understand that Shiloh is not the Prophet who came from his home city called Shiloh, but the split between the Tribes and the secession of the Ten Tribes. That's what Isaiah the Prophet considered the worst thing ever to happen in the History of United Israel until the Ten Tribes were removed from existence. (Isa. 7:17)
Rehoboam, the King who had succeeded Solomon his father did not understand and started preparing the Country for civil war when Shemaiah, the man of God dissuaded him by making him understand that Shiloh had come. He got it and recalled the Army. (I Kings 12:21-24)
Now, kindly share with me your comments.