The Biblical Story of the Israelites
In Genesis 32:22-32, Jacob wrestles with an unnamed man throughout the night. At daybreak, the man tells him, "You shall no longer be called Jacob, but Israel, for you have striven with God and with humans, and have prevailed." "Israel" translates from the Hebrew as "he who strives with God."
Jacob eventually fathers twelve sons by four different women; each of these sons gives his personal name to one of the twelve tribes of Israel. The reader is expected to understand something of each tribe's history and character by following the stories of these boys. (If a modern novelist were to write about two sisters named Virginia and Georgia, along with Virginia's daughter Alexandria and Georgia's daughters Atlanta and Marietta, the impression that the story makes on an American reader might be roughly parallel to an ancient reader's impression of the story of the twelve tribes. If Atlanta and Marietta had different fathers of different social classes, this would add further complexity to the narrative.)
The Israelites, then, are the clan whose story is told in the Tanakh, and who trace their common ancestry through Jacob/Israel back to his grandfather Abraham. Genesis explains that God made a covenant with Abraham, offering him a territory that his many descendants would one day claim for their own. However, through a series of events described in the last few chapters of Genesis and the first few chapters of Exodus, the Israelites found themselves enslaved in Egypt and their dream seemed for a long time unreachable.
Some time later, God reminds Moses of the promise he had made to Abraham. Moses sets out in search of the Promised Land, the newly-freed Israelites accompanying him. Eventually they reach the country called Canaan, and Moses' follower Joshua leads a bloody invasion. Over time, the twelve tribes unite under a single leader and a kingdom is established in Canaan, though within only a few generations it is divided by civil war and is eventually conquered by the armies of neighbouring nations.
In 722 B.C.E., ten of the Israelite tribes were wiped out by the Assyrians, leaving only the two southernmost tribes of Judah and Benjamin. Since Judah possessed a much larger territory than Benjamin, the people who were once called Israelites were soon called Judaean, which in Greek is ioudaios and which gives us the modern word Jew.
In 586 B.C.E., the remaining two tribes were driven into exile by the Babylonians; Jerusalem was razed and the Temple was destroyed. Though the Persian king Cyrus permitted the Jews to return to their homeland a couple of generations later, not all of them chose to do so, and for the rest of their history, the Jews would make their homes all around the world. Some would harbour a hope of returning to the Promised Land, and modern Jews often make the aliyah to the modern state of Israel in order to satisfy this desire. Others, however, are happy where they are, and view the Promised Land as being a place in their hearts and souls rather than a physical country.
Though naming conventions are not absolutely consistent among modern historians, the word Israelite is generally used to describe members of the tribe described in the Tanakh. This term covers patriarchal times, the united and divided kingdoms, and the wars leading up to the Babylonian Exile. Postexilic descendents of the Israelites, by contrast, are referred to as Jews.
Who Were The Israelites, Really?
Modern scholars are divided on the identity of the ancient Israelites. The stories about the conquest of Canaan, told in the Biblical books of Joshua and Judges, are not supported by archaeological evidence. This leads some skeptics to claim that there never was a mass migration into the promised land, and that the authors of the Hexateuch simply described themselves as (former) outsiders without ever having been outsiders.
Some historians go even further with their skepticism, claiming that the kingdoms of David and Solomon never existed, and that the authors of the books of Samuel and Kings simply constructed an elaborate fantasy of cultural supremacy. It does seem rather hard to believe that Solomon could have managed to marry seven hundred women, every single one of them princesses, without earning a single mention in contemporary Egyptian or Mesopotamian sources.
But perhaps it seems even odder to imagine an entire culture inventing a myth about something as humiliating as enslavement by a foreign people and forty years of aimless wandering in the desert. What could have been the inspiration for this story?
Whoever the Israelites were, it seems that YHWH was their tribal god, and that they spent several hundred years in conflict with the polytheistic Canaanites who shared their living space. The transformation of YHWH from a local war-god who lived in a box to the all-powerful creator of the universe is a process that is poorly understood, though surviving Biblical texts do provide some clues about its shape. Even a quick comparison between, e.g., 2 Samuel 6 and Job 38 is enough to demonstrate how the views that the Israelites had of their divine patron changed over time.
Do not confuse Israelites with Israelis. The former term refers to the ancient tribe I have been discussing, while the latter term refers exclusively to citizens of the modern state of Israel, founded in 1948.
Further Reading
Hershel Shanks' Ancient Israel: from Abraham to the Roman Destruction of the Temple is a balanced introduction to both the literary and the archaeological sources for this people's long and complicated history.
Victor Matthews and John Benjamin have assembled a fascinating collection of non-Biblical documents that have relevance for Israelite history in their Old Testament Parallels, though I warn that some of their translations are really dodgy.
And finally, it's always a good idea to read the Bible itself. I prefer the NRSV translation, though the Jewish Publication Society's version of the Tanakh is very fine and can't be accused of having any Christian bias.