Who is an Arab?
It is a Jewish misconception that all Arabs are Children of Ishmael, the son of Prophet Abraham.  It can be said that all of the Children of Ishmael are Arabs, but not all Arabs are Children of Ishmael.  It is also important to remember that not all Arabs are Muslims and not all Muslims are Arab. In reality, only 20% of Muslims are Arabs.  The biggest population of Muslims are actually Indonesians! It is equally wrong to categorize all European Christians as Edomites (from Esau).

There are 3 types of Arabs, they are:

1)  Arab al-Ba'ida - the "lost" Arabs, destroyed for their sins (i.e. 'Ad, Thamud, etc.) who were "hamitic" in description (dark skin, frizzy to kinky hair) but preceded the time of Ham or even Noah (as far as i can tell).  However, many still claim that they were the descendants of Shem as well, which is strange since their appearance is "black" (i.e. Shem, and Semites would be as well) and linguistically is more similar to afro-semitic (i.e. hamitic) language groups (such as berber, ancient egyptian, ethiopian, etc.), and their civilizations have continued uninterupted up to and before the time of Noah.  They can still be found in numerous places throughout Arabia.

2)  Arab al-'Ariba - the "true" Arabs, descended from Qahtan, known as Gahtan in the Tanakh (see Bereshit 10:25-29, Divrei Yamim a 19-23), a descendant of Shem.  Advent of the "Semitic" Arab.  Josephus erraneously identifies his descendants as living on the Cophon river in India (Kesseth Ha-Sofer 123a).  All the other descendants of Qahtan are clearly in southern Arabia, and the tribe of Qahtan is still in existance today stretching from the red sea to the persian gulf.  Although many were lighter in complexion, these Arabs were typically still very dark in complexion.

3)  Arab al-Musta'reba - the "assimilated" Arabs.  Includes all Arabs (linguistically) not mentioned above.  most notably, includes Isma'il, Fihr (Quraish), Bani-Hashim, Ahlul-Bayt, etc.  Isma'il, the son of a non-Arab (Abraham the 'Hebrew') met the tribe of Jurham in Mecca (see 2 above), learnt arabic from them, thus becoming an arab.  He also married 2 women from this tribe, independantly, and his descendants are from them.  The Quraish are understood to be descendants of Qaidar (Kedar in Tanakh) by most accounts (see Bereshit 25:13, Divrei Yamim a 1:29) through Adnan.  Physically, they resemble all peoples since they are an assimilation of all peoples.  There are even oriental looking Arabs who originally come from SE asia or Bukhara.

(Source is quoted from Brother Omar Ha-Redeye)

Who is a Jew?  
When King Solomon (Prophet Sulaiman) died, the Kingdom of Israel split into 2, the northern Kingdom of 10 tribes retaining the name Israel and the southern Kingdom of the tribes of Judah, Simeon, Benjamin, and some Levites going by the name of Kingdom of Judea. When the Bani Israel living in Judea were exiled to Babylon, it was during this time that the Babylonians starting calling the Judeans, Jews for short, and their religion became known as Judaism. The Bani Israel are made up of 12 tribes and Jews primarily make up 4 of those: Judah, Simeon, Benjamin, and Levi.  The Qur'an only makes the distinction between Bani Israel and Jews (Yahud) is when addressing them is the acutal time periods in which they were called by that specific name.  So thusly when Allah is addressing them during the time of Moses they are called Bani Israel because at that time that is what they are called and there were all 12 tribes there.  At the time of Jesus, Allah addresses them as Jews because in that time period that is what they were called.  There is no debate that Jews are Bani Israel.  But not all Bani Israel are Jews. Just so you understand that:  the 4 tribes of the Jews are part of the 12 tribes of Bani Israel.  But not all 12 tribes of the Bani Israel are in the 4 tribes of the Jews.  For more information, refer to my page History of the Hebrews.

So a Jew technically is a descendant of the Judeans (Judea was destroyed by the Romans in 135 CE).  Theologically, a Jew is anyone who is born of a Jewish mother or converts to Judaism.  Spiritually, a Jew is anyone who embraces and lives according to the laws of Moses (the Torah). As the Torah says in Exodus 31:14, "[Therefore] keep the Sabbath as something sacred to you. Anyone doing work [on the Sabbath] shall be cut off spiritually from his people, and therefore, anyone violating it shall be put to death."  And so if they don't follow the laws of the Torah such as the Sabbath, they are breaking the Covenant of Moses and thusly spiritually they are not an Israelite anymore, even though culturally, they remain Jewish.  Just as in Genesis 17:10-14, "Such shall be the covenant between Me and you and your offspring to follow which you shall keep: every male amoung you shall be circumcised.  You shall circumcise the flesh of your foreskin, and that shall be the sign of the covenant between Me and you.  And throughout the generations, every male among you shall be circumcised at the age of eight days.  As for the homeborn slave and the one bought from an outsider who is not of your offsping, they must be circumcised, homeborn, and purchased alike.  Thus shall My covenant be marked in your flesh as an everlasting pact. And if any male who is uncircumcised fails to circumcise the flesh of his foreskin, that person shall be cut off from his kin; he has broken My covenant..." And so if they don't get circumcised, they are breaking the Covenant of Abraham and thusly spiritually they are not an Israelite anymore.  The foundation of Prophet Abraham and Moses's message was the belief in and worship of the One True God.