"Hebrew" is the anglicized form of the Hebrew word 'ibri, which comes from the root word ayin-bet-resh, "to cross or pass over." The word means something along the lines of "ones from the other side/beyond," though the other side of what exactly isn't clear -- perhaps the Jordan River, or perhaps the Euphrates.
The word is used in the Hebrew Bible primarily to differentiate the group of people who would come to be called Israelites from other people-groups, such as Egyptians or the Philistines. Since the word "Israelite" (in biblical Hebrew, ben yisrael) would be sufficient to denote ethnic difference, it may be the case that the term 'ibri additionally signifies a class status, since it usually occurs in texts where the people so named are enslaved.
When the people of Israel are not experiencing slavery in the Hebrew Bible, the most common term used is b'ney yisrael(literally, "sons/children of Israel").
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