Historians and scholars are in general agreement that Jesus (Yeshua) spoke Aramaic the traditional language of Judea in the first century AD. The Aramaic spoken was most likely a Galilean accent different from that of Jerusalem. Jesus (Yeshua) spent most of his time Nazareth and Capernaum in Galilee, which were Aramaic-speaking villages. The Gospels show Jesus using various Aramaic terms: talitha koum (Mark 5:41); ephphatha (Mark 7:34); eloi eloi lama sabachthani (Matthew 27:46; Mark 15:34); abba (Mark 14:36) to name a few.
Aramaic was similar to Hebrew, but with much of its semantics derived from other languages and cultures, most notably Babylonian. The Dead Sea scrolls indicate that Aramaic remained the principle language of the Hebrews until the Bar Kokhba revolt after which Hebrew became the dominate language. The cosmopolitan nature of the region at that time allowed many languages to mix and intertwine.
During the time of Jesus (Yeshua) the culture was multi-lingual in which Aramaic, Koine Greek and Hebrew were spoken by many. Under Roman rule many Greek speaking Gentiles lived in the cities around Nazareth. Scriptures show that Jesus spoke to the Roman Centurion (Matt. 8:5-13) and to Pontius Pilate (John 18:28-38). It is unlikely that these two men, gentiles knew Aramaic so Jesus (Yeshua) would have spoken to them in Koine Greek or Latin.
Three millennia after it emerged, Aramaic survives today as Neo-Aramaic, which includes different dialects of the religions and the surrounding region. As native speakers have migrated out of the region the language is less and less used and faces possible extinction in the future.
So, listen to this two-minute video of a Christmas Charity Concert at Campo Santo (Dec. 2016), the Vatican. Sarah Ego (Eliyo) is singing the "Our Father / Pater noster" in Aramaic, the mother tongue of Jesus.