Okay, so the Bible identifies Abraham as the first Hebrew ("Ivri" in the original). In the modern sense, "Jew" and "Hebrew" are theoretically interchangeable, although "Hebrew" nowadays is primarily used to refer to the language, rather than the people who speak it.
G-d promised Abraham that his progeny would become a great nation, and that all the nations of the world would be blessed through his descendants.
The recipient of Abraham's spiritual heritage was Isaac. Isaac passed it on to Jacob. Jacob was given the name "Israel," after the incident where he wrestled with an angel (check your Bible). The children of Jacob are Israelites. Ten of the Israelite tribes were exiled when the Assyrians invaded, and various theories exist as to where these "Lost Tribes" wound up. The Jews of today are the spiritual, and in most cases biological, descendants of Israel. And most of them are descended from Israel's son Judah (Yehudah). That's where "Jew" (or "Yehudi" in other languages) comes from.
Not all Jews today are Judean, however. Some are Benjaminites (the Southern Kingdom of Judah, that wasn't exiled until much later and whose descendants largely remained Jewish, consisted primarily of the tribes of Judah and Benjamin). Others are Levites (tribe of Levi), and others still are specifically descendants of Aaron, and are called Kohanim ("Priests"). Because specific tribal affiliation passed through the father (unlike Jewishness itself, which comes from the mother), one can sometimes deduce a person's tribal affiliation through their last name. Levi'im (Levites) often have surnames like "Levi," "Levy," "Lewin," etc. Common Kohein surnames include "Cohen," "Kahana," and "Katz" (this last one is an abbreviation for "Kohein TZedek" or "Righteous Priest").
Whether specifically from Judah or not, all of the above groups are referred to as "Jews". And it makes sense even to refer to Abraham, Isaac, and Jacob as the first Jews even though they came before Judah, because they are the patriarchs and earliest practitioners of the religion that would later come to be called Judaism. However, you could not call them "Israelites," because they are not children of Israel (that last one IS Israel).
I hope this answers your question - for further reading, please check my list of answers. Several of them address this very same question, and the others you might also find of interest. If you have any further questions, feel free to contact me directly.