From The Hebrew Union College Jewish Institute of Religion:
Biblical Aramaic
Aramaic displaced Hebrew for many purposes among the Jews, a fact reflected in the Bible, where portions of Ezra and Daniel are in Aramaic. Some of the best known stories in biblical literature, including that of Belshazzar's feast with the famous "handwriting on the wall" are in Aramaic.
Jewish Aramaic Literature
Aramaic remained a dominant language for Jewish worship, scholarship, and everyday life for centuries in both the land of Israel and in the diaspora, especially in Babylon.
Among the Dead Sea Scrolls, the remains of the library of a Jewish sect from around the turn of the Era, are many compositions in Aramaic. These new texts also provide the best evidence for Palestinian Aramaic of the sort used by Jesus and his disciples.
Since the Jews spoke Aramaic, and knowledge of Hebrew was no longer widespread, the practice arose in the synagogue of providing the reading of the sacred Hebrew scriptures with an Aramaic translation or paraphrase, a "Targum" In the course of time a whole array of targums for the Law and other parts of the Bible were composed. More than translations, they incorporated much of traditional Jewish scriptural interpretation.
In their academies the rabbis and their disciples transmitted, commented, and debated Jewish law; the records of their deliberations constitute the two talmuds: that of the land of Israel and the much larger Babylonian Talmud. Although the talmuds contain much material in Hebrew, the basic language of these vast compilations is Aramaic (in Western and Eastern dialects).
Christian Aramaic Literature
Although Jesus spoke Aramaic, the Gospels are in Greek, and only rarely quote actual Aramaic words. Reconstruction of the Aramaic background of the Gospels remains a fascinating, but inordinately difficult area of modem scholarly research.
Christians in Palestine eventually rendered portions of Christian Scripture into their dialect of Aramaic; these translations and related writings constitute "Christian Palestinian Aramaic".
A much larger body of Christian Aramaic is known as Syriac. Indeed, Syriac writings surpass in quantity all other Aramaic combined. Syriac is originally the literary language of the city of Edessa (now Urfa in SE Turkey). The language became the tongue of the entire eastern wing of the church, from about the third century C.E. down until well past the Muslim conquest.
Syriac writings include numerous Bible translations, the most important being the so-called Peshitta (simple) translation, and countless devotional, dogmatic, exegetical, liturgical, and historical works. Almost all of the Greek philosophical and scientific tradition was eventually translated into Syriac, and it was through this channel that most found their way into the Islamic World and thence, into post-Dark Ages Europe.
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http://cal1.cn.huc.edu/aramaic_language.html
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http://www.huc.edu/faculty/faculty/index.shtml