Jesus' spoke Aramaic, the common language of Galilee during his lifetime. Aramaic was an ancient Semitic language related to Hebrew much as French is related to Spanish or as Cantonese is related to Mandarin. (Thanks to Prof. Zev bar-Lev for help with these analogies.) Though Jews had once spoken Hebrew as their primary language, this changed when Israel was overthrown, first by the Assyrians in the eight-century B.C. and then by the Babylonians in the sixth-century B.C. By the time of Jesus Aramaic was so common among Jews that the reading of the Hebrew Scripture in the synagogue was accompanied by translation into Aramaic. (For a helpful overview of Aramaic, see the "Comprehensive Aramaic Lexicon" website of Hebrew Union College.)
In addition to the strong circumstantial evidence that Jesus spoke Aramaic as his primary language, we find direct evidence for this theory from the New Testament gospels. Though these gospels were written originally in Greek, at several points Jesus' words are given in Aramaic, for example: "Talitha cum" (Mark 5:41, "Little girl, get up!"); "Abba" (Mark 14:36. "Father"); "Eloi, Eloi, lema sabachtani?" (Mark 15:34, "My God, my God, why have you forsaken me?"). In these cases the actual Aramaic words of Jesus were remembered and passed on even by Greek-speaking Christians.
These passages and others from the gospels, combined with the predominance of Aramaic in Palestine in the first century A.D., make it virtually certain that Aramaic was Jesus' primary language. I don't know of any serious scholar who doubts this conclusion.