Aramaic is very similar to Hebrew and was the common language of Israel in Jesus' day. After the Jews returned from the Babylonian Captivity Aramaic had gradually replaced Hebrew as the ordinary speech of the Jewish people. But in Jerusalem the rabbis spoke Hebrew, and the commentaries on the Torah were written in Hebrew. The Essenes also spoke Hebrew, we know this because when the Dead Sea Scrolls were discovered the majority of their writings were in Hebrew. It is interesting that in New Testament times the actual words recorded were in Aramaic, and not in Hebrew.
Aramaic is a Semitic language, and was the ancient language the Arameans of Syria, a dialect very similar to Hebrew. The Assyrian king Tiglath Pilesar I (1110 BC) was the first to mention the Arameans of Syria Anatolia. Aramaic was common throughout the ancient near East because it's alphabet was much easier than cuneiform which was used by the Assyrians and the Persians. Aramaic was actually used by the Assyrians and the Persians for diplomatic and international relations. Scholars refer to Aramaic from the Assyrians to the Persians to the Greeks as Imperial Aramaic, a uniform script found on many inscriptions throughout the ancient near East from Anatolia to Persia and Afghanistan. Before that it was referred to by scholars as Old Aramaic.
The Old Testament Books and Aramaic
Some of the books in the Old Testament contained Aramaic. Ezra 4:8 - 6:18, Ezra 7:12-26, Daniel 2:4b-7:28, and one verse of Jeremiah (Jer 10:11) were written in old Aramaic.
The New Testament Books and Aramaic
Scholars refer to Middle Aramaic as the period from the Maccabean's to about 200 A.D. which was replaced by the Greek language. Middle Aramaic can be found in New Testament writings, some of the Dead Sea scrolls, the texts of Bar Khochba, the Nabateans, and others. When Jesus cried out from the cross "Elo-i, Elo-i lama sabach thani" (Mark 15:34) he was quoting Aramaic. Other words in the New Testament like Talitha cumi (Mark 5:41), Peter's name Cephas is from an Aramaic word "kepha" meaning rock. Thomas is from the Aramaic word "toma" which means twin, and the word "bar" in names like Bartholomew, Barabbas, and Bartimaeus is the Aramaic word for son. The Hebrew word for son is "ben". Golgotha is the Aramaic word for skull, and maranatha (1 Cor 16:22) is from two Aramaic words maran (our Lord) and eta (come).
Aside from the New Testament, Dead Sea Scrolls, and Bar Kochba texts, the only other surviving document from Israel in Aramaic is the Megillat Tacanit, a rabbinic scroll of fasting. The remaining Aramaic texts are found on inscriptions on things like limestone ossuary's, although some of the ossuary's were written in Hebrew. There are also the Targums which were commentaries of the Old Testament written in Aramaic. The most common on the writings of Moses is known as the Onkelos" which was a very literal translation in Aramaic found among the Dead Sea scrolls.