Question: "What language did Jesus speak?"
Answer: While Jesus very likely spoke Aramaic, Hebrew, and Greek, Aramaic was likely the language Jesus spoke the most. The Gospels record Jesus speaking numerous Aramaic words: talitha koum (Mark 5:41); ephphatha (Mark 7:34); eloi eloi lama sabachthani (Matthew 27:46; Mark 15:34); abba (Mark 14:36). Historians, archaeologists, and cultural anthropologists are almost universally agreed that Aramaic was the common or colloquial language in Israel during Jesus’ time. Aramaic was very similar to Hebrew, but with many words and phrases that were borrowed from other languages and cultures, especially Babylonian.
Hebrew was spoken primarily by the scribes, teachers of the law, Pharisees, and Sadducees, the “religious elite.” Hebrew was likely often read in the synagogues, so most people were probably able to speak and understand some Hebrew. Since Greek was the language of the Romans, who had power over Israel during Jesus’ time, Greek was the language of the political class and anyone who wanted to do business with the Romans. Greek was the universal language at that time, so, the ability to speak Greek was a highly desirable skill. Some, however, refused to speak Greek out of resentment toward their Roman oppressors. When Jesus spoke with Pontius Pilate, it is possible that He spoke to him in Greek, although Pilate, as the governor, likely would have been able to speak Aramaic as well.
Jesus, as God incarnated in human form, could have spoken any language He chose. In His humanity, Jesus likely limited Himself to the languages common to His culture: Aramaic, Hebrew, and Greek. Jesus likely spoke whichever of the three languages was most appropriate to the audience He was addressing.