Aramaic is a Semitic language related to Hebrew in much the same way as Spanish is related to Italian. It was a major language in the Middle East and beyond in the centuries before and after the time of Christ. It seems to have been (in varying dialects) the language of the common people throughout much of the Middles East at the time of Christ.
Although the Jews of the time must have known and spoken Hebrew, it is still likely that the common language of the Bible lands was Aramaic. Of course, they may have simply considered it a different form of Hebrew in the same way that the Arabic of Morocco cannot be understood by the Arabs of Iraq and vice versa, but they all consider themselves to be speaking Arabic. I say this because Aramaic is not mentioned as a distinct language in the New Testament. Therefore, it may have been considered as another form of Hebrew because of its similarity.
The evidence for Jesus speaking Aramaic is found in some of the statements in the gospels that are transliterated in the King James Bible. For instance:
Mark 5:41 And he took the damsel by the hand, and said unto her, Talitha cumi; which is, being interpreted, Damsel, I say unto thee, arise.
The statement is said to be Aramaic and not Hebrew or Greek. Also notice that it is translated for those who were reading it in Greek. Another example is:
Matthew 27:46 And about the ninth hour Jesus cried with a loud voice, saying, Eli, Eli, lama sabachthani? that is to say, My God, my God, why hast thou forsaken me?
This statement is also said to be in Aramaic. As I said before, Aramaic is closely related to Hebrew but it is not the same language. Therefore, whenever someone was speaking in Aramaic in the Bible, this had to be translated into Greek before it was put in the New Testament books as Greek. However, this pretranslation would have been required whether they spoke Aramaic or Hebrew (for another example of pretranslation see Acts 21:40 where Paul spoke in Hebrew but the book of Acts was written in Greek). Greek was the universal language, Latin was the governmental language, Hebrew was the religious language of the Jews, and Aramaic was the common language of the Middle East. These folks were quite linguistic no matter how you look at it.
However, there is another teaching behind this claim. Some teach that the books of the Bible were originally written in Aramaic and then had to be translated into Greek. The only problem is that there is no proof of this. Both Matthew and Hebrews have been taught to have been originally written in Hebrew or Aramaic. And, although, it is possible that they might have been written in Hebrew, the evidence is lacking. This is conjecture at its best--or worst. To use this shot in the dark to prove anything about Bible translation is less than weak; it is foolish.
One thing that is accepted is that the following Old Testament passages are in Aramaic: Daniel 2:4-7:8; Ezra 4:8-6:18; 7:12-26; Jeremiah 10:11. This almost certainly goes back to the Persian influence in language at the times these books were written.
As a final thought, I would like to mention that there are numerous places in the Bible where a speech or statement is recorded in a language other than the one in which it was originally made. For instance, it is highly unlikely that the Pharaoh of Egypt spoke to Moses and Aaron in Hebrew. Therefore, his words as found in the Bible were translated before they were recorded. There are many other instances of this in the Bible.
Therefore, God had to be in charge of translation in order for the words of scripture to be perfect in the original copies. It seems strange that people cannot trust Him to work in translation today. Thanks for the question.