Tribes of Israel
Israel was the name that God gave Jacob in the book of Genesis. Jacob had twelve sons (by his two wives, Leah and Rachel, and their handmaids, Bilhah and Zilpah), each of which was traditionally the ancestor of a tribe of Israel. These sons were:
Although the tribes are often called the "twelve tribes of Israel", there are actually thirteen. Joseph had two sons, Manasseh, the firstborn, who had his own namesake tribe, and Ephraim, the whose descendants were the tribe of Joseph. Nevertheless, in most lists in the Bible, only twelve tribes are listed.
Of the thirteen tribes, each was allocated its own land ("inheritance") after the exodus, except Levi, which was a scattered tribe given several religious duties. The tribe of Simeon dwindled in size and was eventually assimilated into the tribe of Judah.
After the reign of Solomon, most of the tribes of Israel rebelled against the leadership of the tribe of Judah, leaving only Judah (including Simeon), Benjamin, and some priests from the tribe of Levi under the leadership of the king at Jerusalem. The Bible eventually loses track of the tribes that rebelled, after both Israel and Judah are conquered by the Assyrians. Oddly, it counts these "lost tribes" as ten, even though it explicitly states that Judah, Benjamin, and part of Levi remained one nation until that point, and the tribe of Simeon was said to have originally resided entirely within the nation of Judah.
Nonetheless, the Bible still contains prophecies about these tribes, and a great deal of theological speculation has sprung up over what happened to these tribes (the historical record seems to suggest that the nation of Israel was simply assimilated into other nations after the Assyrian conquest). Mormons believe that these tribes were scattered (with some individuals even ending up among the Native Americans), and seek to reunite the tribes of Israel.
The Jewish people claim to be descended (largely) from the nation of Judah. Membership in particular tribes has been forgotten/eliminated, except that some Jewish traditions still give the Levites particular religious duties.