The Targums are the Aramaic translations--many authors say paraphrases--of  the books of the Old Testament.  In their earliest form, they date from  the time when Aramaic was superseding Hebrew as the spoken language of the Jews.   They were designed to meet the needs of the unlearned among the  people who had ceased to understand the Hebrew of the Old Testament. 

Jewish tradition places the beginning "targums" in the days of Nehemiah  when Judah is returning from the exile.   We read in Nehemiah 8.8 "They read from the Book of the Law of God, making  it clear (translating it) and giving the meaning so that the people could  understand what was being read."   It is easy to think that many of  the returning exiles had lost more than a little facility in Hebrew during the  exile.  Especially since many did not return at all.  We also know  from the story that those that did return had lost much of their knowledge of  what was contained in the law.  The written Targums that have come down to us are of much later  date. 

The custom among the Jews today is to read a Torah Parsha and a Haphtarah reading in the weekly  synagogue meetings.  In the meetings I have attended the readings are  followed by an English translation.  As we know from the New Testament, the custom of reading in the synagogues both  from the Law (Acts 15:21) and from the Prophets (Luke 4:16 f.; Acts 13:14, 27) was well  established in the 1st century A.D.  The practice of accompanying these readings with a  translation into Aramaic is, further, so generally recognized by the 2nd century  A.D. that the Mishna takes it for granted, and merely inculcates  certain regulations to be observed by the Meturgeman (translator), who  had by this time acquired a definite status.

From the Mishna we learn that the Meturgeman, who was distinct from the reader, translated each verse of the  Law into Aramaic as soon as it had been read in Hebrew: in the readings from  "the Prophets" three verses might be read at a time. Later regulations are also  laid down in the Talmuds in order to prevent any appearance of authority  attaching to the translation, and also to ensure reverential treatment on the  part of the translator. The pattern of reverance continues to this day, but the  regulation are more lax.  Of course we all sit with copies of the Tanakh in our laps that contain both the English  and the Hebrew.

Based in large part on John F. Stenning, Encyclop√¶dia Britannica, 11th ed. (1911)  on http://www.bible-researcher.com/aramaic3.html accessed 9/9/06