Adonai means LORD and was often used as a substitute for God's name "Yahweh" or "Jehovah" (these are interchangeable) in the Old Testament.
The Hebrews felt that God's name was too sacred to pronounce so they eliminated the vowels and used YHWH which was unpronounceable. Still, they had to be clear about whom they meant so they often spoke of Adonai - the LORD who is GOD. If this name was used to mean someone other than God, "Adoni" was used which was a common word for one who was a master and ruler.
Psalm 110:1 provides a good starting point to understand who "Adonai" is:
"The Lord (YHWH) said to my Lord (Adonai) 'Sit at My right hand, till I make Your enemies Your footstool."
This verse is commonly understood by both Jews and Christians to be prophetically speaking of the reign of the Messiah. The psalm is written by King David who is saying, "The God of the universe said to my (meaning David's) Lord the Messiah…"
The promise from God is that the Messiah will sit at His right hand and that His enemies would become His footstool. Does that fit the bill for Jesus? Jesus discussed this passage in Luke 20:41-44 with the scribes (religious leaders of the times) and tried to get them thinking about who the Messiah might be. He was shaking up all their pre-conceived ideas and trying to get them to think "outside the box" as we would say today.
In the Gospel of Mark 14:61-63, Jesus speaks more plainly. Before he was crucified, Jesus had to appear before the high priest and this conversation took place:
Again the high priest asked Him, saying to Him, "Are You the Christ, the Son of the Blessed?" Jesus said, "I am. And you will see the Son of Man sitting at the right hand of the Power, and coming with the clouds of heaven." Then the high priest tore his clothes and said, "What further need do we have of witnesses? You have heard the blasphemy!"
The high priest called it blasphemy because he was aware that Jesus was calling himself the Messiah and claiming that he was the one spoken of in Psalm 110. And so they went ahead and condemned him to death.
Later, on the Day of Pentecost when the Holy Spirit fell, after Jesus had already ascended into heaven, Peter preached from Psalm 110 one more time. After he quoted the verse he said; "Therefore let all the house of Israel know assuredly that God has made this Jesus, whom you crucified, both Lord and Christ." In this case the hearers did not reject the message and about 3000 people were converted.
Today, Jesus sits at the right hand of the Father in heaven. Through his death and resurrection he completely spoiled his enemies and they are a footstool under his feet just as YHWH promised in Psalm 110.
Jesus Christ is Master, Ruler. He is Adonai.