A set of passages usually called the Servant Songs, Isaiah 53 chief among them, describe a figure who is despised and rejected, "wounded for our transgressions, crushed for our iniquities," whose suffering is somehow borne on behalf of others. The earliest Christian writing already applies this directly to Jesus: Acts has Philip explain the passage to an Ethiopian official by preaching Jesus from it, and the reading of Isaiah 53 as a messianic prophecy fulfilled in the crucifixion has remained continuous in Christian tradition since.
The dominant reading in rabbinic Judaism identifies the servant collectively as the nation of Israel, suffering among the nations for its faithfulness, an interpretation with roots stretching back near the origin of the debate itself. It is not, however, the only Jewish reading on record: a minority of medieval and modern Jewish commentators, among them the fourteenth century Rabbi Moshe Kohen ibn Crispin of Cordova, argued for an individual referent, though not for Jesus specifically, and criticised the corporate reading as a departure from the passage's plain sense. Both the collective and the individual non-Christian readings agree in rejecting the specifically Christian identification, while disagreeing with each other about what the alternative is.
Sources The Hebrew Bible (Tanakh)Anonymous (multiple authors) with The New Testament, Acts of the ApostlesAnonymous