Aramaic has been in some ways a forgotten language in biblical studies, except at a very high academic level. The New Testament is written in Greek; nearly all the Old Testament is written in Hebrew, while the Greek translation of the Old Testament (the LXX) is significant to biblical studies. Yet 268 verses of the Bible were written in a language called Aramaic.
The portions of Scripture that were written in Aramaic include Ezra 4:8–6:18 and 7:12-26 (67 verses), Daniel 2:4b–7:28 (200 verses), Jeremiah 10:11, and various proper names and single words and phrases scattered throughout the Old and New Testaments. Despite the relatively small percentage of Scripture that is written in this language, the Aramaic portion of the Bible is disproportionately significant because of the importance of the book of Daniel to biblical prophecy. Aramaic is also important for New Testament studies, as several direct quotes from Jesus and others are preserved in the original Aramaic that was spoken by Palestinian Jews of the Second Temple period. New Testament verses which include Aramaic words transliterated by Greek letters are: Matt 5:22; 27:46; Mark 5:41; 7:34; 10:51; 14:36; John 1:42; 20:16; Acts 9:36, 40; Rom 8:15; 1 Cor 16:22; Gal 4:6.
In the Old Testament, four verses make a direct reference to the Aramaic language: 2 Kings 18:26, Ezra 4:7, Isaiah 36:11, and Daniel 2:4. Each of these verses calls Aramaic “Aramaic” (אֲרָמִית, an adverbial form of אֲרָמִי), though this used to be translated as “Syrian” or “Chaldee” in English. Aramaic is called “Hebrew” (Ἑβραΐς or Ἑβραϊστί) in the New Testament, since it was the tongue of the Hebrews (John 5:2; 19:13, 17, 20; 20:16; Acts 21:40; 22:2; 26:14). Some newer translations render the Greek word for “Hebrew” in these verses as “Aramaic,” which recognizes that these verses refer to the language we now call Aramaic.
Aramaic was originally the language of the Arameans, who were comprised of tribes that lived along the Euphrates River. Two of the most prominent of these tribes were the Syrians to the northwest, and the Chaldeans to the southeast. The word Aramaic is derived from Aram, a son of Shem who was the progenitor of the Arameans. In the earliest stages of the history of Aramaic, the language was only spoken in Aramean locales, including the area where Laban lived (cf. Gen 31:47; Deut 26:5). However, as the Syrians and Chaldeans gained prominence in the ancient Near East, their tongue became established as an international language of commerce and diplomacy, gradually displacing Akkadian. Akkadian was still the official language of the Neo-Assyrian Empire, although 2 Kings 18:26 indicates that Aramaic was already becoming established as a lingua franca of the ancient Near East by 700 BC. When the Chaldeans subsequently conquered Assyria, it was natural for them to use their own language of Aramaic as the administrative language of the Neo-Babylonian Empire, rather than adopting Akkadian. This is why Daniel 2:4 says the wise men of Babylon addressed the king in Aramaic, and why the following section of the book of Daniel is written in Aramaic. After the conquest of Babylon by Persia, the Persians also established Aramaic as the official language of their vast empire. This is why the portions of Ezra which record official correspondence are written in Aramaic.
At the time when the books of Daniel and Ezra were written, most Jews could speak and understand both Hebrew and Aramaic. They understood Hebrew as the language spoken at home, among themselves, and in the reading of the Scriptures, while Aramaic was the language spoken in broader society. Over time, Aramaic replaced Hebrew as the primary language spoken by the Jews who lived in Palestine and regions to the east. The Jews had not learned Aramaic in Palestine (cf. 2 Kgs 18:26), but they had to learn it in exile, since it was the language of their captors. Thus, the parts of the Old Testament which were composed in Aramaic were written in that language as a result of the Babylonian captivity.
Because of this, Aramaic was the native tongue of our Lord; Hebrew was rarely used as a spoken language by Jews of the first century AD. There are several places where the Gospel writers preserve quotations from Jesus in the original Aramaic, including His cry from the cross, Eloi, Eloi, lama sabacthani? (Mark 15:34). These words expressed Christ’s deepest feelings at a time of great personal anguish and emotion. That He spoke these words from Psalm 22:1 in Aramaic, rather than from the Hebrew original or the Greek Septuagint translation, shows that Aramaic was the language that He knew most intimately. Thus, the New Testament preserves Aramaic words because Aramaic was the mother tongue of Palestinian Jews in the first century AD.
The Greek of the New Testament was influenced by Aramaic, and so contains some Aramaic idioms and forms of expression, such as the phrase “answered and said.” Although the degree of Aramaic influence on the Greek of the New Testament has been a subject of much debate, it is fair to say that the style of New Testament Greek is Semiticized to one degree or another. But it is not true that parts of the New Testament were originally written in Aramaic, as some have claimed. No manuscript of any part of the New Testament has ever been discovered that is written in the Jewish Palestinian Aramaic dialect known to Jesus and the apostles.
After the resurrection of Jesus, the Syriac dialect of Aramaic became the language of the Syrian church. Aramaic also remained an important language for the Jews. Because of this, there are two major Aramaic translations of the Old Testament, the Jewish Targums and the Syriac Peshitta. There are a number of important Syriac versions of the New Testament. Much of Jewish rabbinic literature, and nearly all Syrian Christian literature, is written in Aramaic. Some of the Dead Sea Scrolls were also written in Aramaic.
Both Aramaic and Hebrew are West Semitic languages. Thus, Aramaic and Hebrew share many of the same linguistic characteristics and modes of expression. Overall, Hebrew grammar and morphology is somewhat closer to proto-Semitic, especially in its patterns of vocalization, though Aramaic has a fuller complement of distinct verbal stems. Some distinctive characteristics of Aramaic include the frequent use of the participle for a finite verb, the versatile particle דִּי, the use of a determined form instead of a prefixed definite article, and such idioms as “son of man” (for “man”) and “answered and said” (for “said”). Because of the importance of Aramaic in the Second Temple period, Hebrew gradually began to be written in Aramaic letters during that time, and Hebrew has used the Aramaic square script ever since. However, Syriac and other dialects of Aramaic use different scripts, while the Targumim have a system of pointing that differs from the Masoretic pointing of the Old Testament.
One of the peculiarities of biblical Aramaic is that the divine name יהוה (Yahweh) is never used. For some reason, this name was only used in Hebrew. However, the term אֱלָהּ שְׁמַיָּא (the God of heaven) occurs very frequently in Aramaic, much more than in Hebrew. It is also interesting that there are no Old Testament books written entirely in Aramaic. This is apparently to retain the character of the Old Testament as a Hebrew text.
Because of the very long linguistic history of Aramaic, and the diverse number of groups that have spoken it, there are quite a variety of Aramaic dialects, of which Syriac is the most prominent. Some eighty percent of extant Aramaic writing is in Syriac, a language which is still spoken today (in various dialects) and is used in the liturgy of some Eastern churches. There are also distinct differences between different chronological periods of Aramaic. Although liberal scholars have long attempted to deny it, the Aramaic of both Daniel and Ezra is of the Imperial Aramaic dialect that would have been in use in the sixth century BC. It is noticeably different from both the Aramaic of Qumran and from first-century AD Jewish Palestinian Aramaic.
While Hebrew was used sparingly outside of the Bible, Aramaic was used very broadly. There is a huge corpus of Aramaic literature. From about 600 BC until AD 700, Aramaic was the primary trade language of the ancient Near East. It was also the primary spoken language of Palestine, Syria, and Mesopotamia at the time of Christ. Aramaic was only displaced by Arabic when the Muslims conquered the Middle East—though the language never died out completely, and is still spoken in pockets of Syria, Iraq, Iran, and Turkey. Aramaic is possibly the language with the longest continuous written record in the world. Because of the broad use of Aramaic outside of the Bible, there is rarely any doubt about the meaning of words or constructions in biblical Aramaic, as there are many opportunities to research their usage in extrabiblical literature.
Although there is only a limited amount of biblical material composed in Aramaic, the influence of the Aramaic language is felt throughout the Old and New Testaments, as it was present in the background from Genesis until Revelation. Aramaic also had a prominent place in the early church and in postbiblical Judaism. But insofar as it is directly used in the Bible, Aramaic is the language of the captivity and of the Redeemer.
Postscript: For recommended resources for the study of Biblical Aramaic, see this post.
 Since the discovery of the Dead Sea Scrolls, there has been a debate over the extent to which Aramaic had displaced Hebrew in Palestine by the first century AD. However, first-century AD inscriptions in Palestine are almost exclusively in Aramaic (or Greek), and Aramaic is consistently used by Jesus, rather than Hebrew. Jesus probably understood Hebrew, but as a literary, rather than spoken, language. He would have known Greek as a second language and spoke it on some occasions (as when dealing with Gentiles), but He would have been more at home in Aramaic.
 The Aramaic square script is also called the “Jewish script,” the “square script,” or the “Assyrian script.” Three stages in the development of this script at Qumran are called the “archaic script” (250-150 B.C.), the “Hasmonean script” (150-30 B.C.), and the “Herodian script” (30 B.C. – 70 A.D.). Despite the prevalence of the square script in Hebrew writing, twelve Qumran fragments were found written in a paleo-Hebrew script similar to the original Hebrew script in which most of the Old Testament was written, while several other Qumran manuscripts used the square script for the main body text and the paleo-Hebrew script for nomina sacra. See E. Tov, Textual Criticism of the Hebrew Bible (3rd ed.; Minneapolis: Fortress Press, 2012), 206-7.