In my opinion, the central concept of a religion is what distinguishes it from other religions. Christianity and Islam both believe in monotheism just as Judaism does. So I feel that what distinguishes Judaism from other religions is the belief that Jews have a special relationship with God. This is distinct from the universalism of Christianity, Islam, and Buddhism.
I think that that collective nature is another thing that differentiates Judaism. When a Christian sins, he’s punished; when a Jew sins, all Jews are punished. The covenant is between God and the Jewish people as a whole, and not between God and any individual Jew.
They are God’s chosen people.
I disagree. The central concept is something central the religion itself, not just a field mark to distinguish it from other religions.
Only because Christianity and Islam started with Judaism and added stuff. It seems unfair to say that a religion has lost its “central concept” just because someone else starts a competing religion that steals the concept but denigrates it to second place behind some other concept.
I don’t know Islam well enough to judge what is central to it, but “there is exactly one God” may be it’s central concept, too.
Surely a central concept of Judaism is that, no, Christianity doesn’t believe in monotheism the way Judaism does. (Or: the central concept of Christianity is monotheism, feh.)
There are many things that distinguish Judaism from other religions, but that doesn’t make any of them the central concept. You may as well say that observing the Sabbath on Saturday is the central concept of Judaism.
I’ve heard it expressed that the genius of Judaism is to proclaim that God is good. The genius of Islam is to proclaim that God is great. And the genius of Christianity is to proclaim that God is small.
I believe that "G-d is great, G-d is good is common to all People of the Book.
G-d is small? Please explain. 
This sorta comes from C.S. Lewis, who I read extensively when I was a teenager, and so of course is from a theistic perspective I no longer share; but the idea is that Judaism perhaps introduced, but certainly stressed, that God is an ethical, moral God, that he does what is righteous, and indeed is the source and font of all righteousness. As other posters have noted, the God of the Jews is concerned with justice and right behavior, especially as practiced by his followers.
Islam stresses that God is vastly greater and fundamentally different than anything human. He is the ultimate source of peace, and the ultimate source of mercy, and the ultimate source of justice. He is utterly unhuman, so worship of anything other than Allah - an image, a saint, a desire - is a desecration of the only being in creation that can be called a god.
Christianity says that this all-powerful, all-knowing and all-merciful god was so loving of his creation that he humbled himself to become the weakest thing in the world: the bastard child of an unwed teenager. Who grew up to die a disgraceful and horrible death on the cross, just so that as he became Son of Man, we could become sons and daughters of God. He would know suffering just as we know suffering; and his suffering would redeem ours.
G.K. Chesterton makes this point at the climax of his allegory The Man Who Was Thursday. All the men have gathered with God to complain of the torments he put them through, when Satan comes in as well, to accuse them of being safe, powerful and indifferent:
Syme, the protagonist of the novel, springs to his feet to passionately rebut the charge:
He turns his face to God, to ask if he had ever suffered:
That’s what I think is meant by the saying that the genius of Christianity is to proclaim that God is small. The unique message of Christianity is that God became human. It’s why that even though I no longer accept it as a truth, the Gospel story still moves me.
Mods, The Man Who Was Thursday is in public domain, which is why I broke the rule on quoting from a published work.