The World as Reward
Lucifer, a spirit being, was specially created and consecrated by God to serve an exalted position at his throne as a covering cherub (Ezekiel 28:14). His service, over time, was so extraordinary that he became the first (and only known) angelic being to be rewarded with a responsibility to rule part of God's creation.
God promoted Lucifer by sending him to earth with authority over both the world and one-third of the angels (Revelation 12:4). When Lucifer sinned he was condemned for trying to exalt his throne above all that existed including his Creator!
How you are fallen from the heavens, O shining star, son of the morning! . . . For you have said in your heart, 'I will ascend into the heavens (from the world), I will exalt MY THRONE above the stars of God (angels) . . . I will be like the Most High.' (Isaiah 14:12 - 14, HBFV).
God allowed Lucifer to retain his rule over the world even after he sinned. It was this authority he would later try to leverage in his attempt to get Jesus to sin.
A Bold Assertion
Jesus, at the start of his ministry, had to first endure 40 days of whatever temptation Satan the devil decided to throw at him (Matthew 4, Mark 1, Luke 4). The last, and by far the greatest, temptation that came his way included an amazingly bold assertion.
Then the devil took Jesus and in a moment of time showed him all the kingdoms of the world.
The devil said to him, "I will make you king over all these places. You will have power over them, and you will get all the glory. IT HAS ALL BEEN GIVEN TO ME. I can give it to anyone I want . . ." (Luke 4:5 - 6, ETRV).
Jesus' response rejected the devil's demand that he first commit idolatry before receiving what was promised. He did not, however, refute the devil's claim of authority over the world.
"Satan showed Jesus the kingdoms of the world with all their splendor. These kingdoms presently are Satan's . . . He had the power to give all these kingdoms to Jesus at that time . . ." (Bible Knowledge Commentary).
A Transfer of Authority
Jesus, on the Saturday before his crucifixion, revealed to his disciples that his death would annul the devil's authority.
"Now is the judgment of this world. Now shall the prince of this world be cast out. And if I be lifted up from the earth, I will draw all to Myself" (John 12:31 - 32, HBFV, see also 1John 3:8).
Jesus' sinless sacrifice qualified him to replace Satan's power over the world. His triumph over the powers of darkness was revealed to his disciples, in Galilee, shortly after his resurrection.
"All authority in heaven AND ON EARTH has been given to Me." (Matthew 28:18, HBFV).
A Contradiction?
If the devil's authority over the world was given to Jesus then how can we explain the following statements made years after his resurrection?
Because we (Christians) are not wrestling against flesh and blood, but against principalities and against powers, against the WORLD RULERS of the darkness of this age . . . (Ephesians 6:12, HBFV).
But if our gospel is hidden, it is hidden to those who are perishing; In whom THE GOD OF THIS AGE (the devil) has blinded the minds of those who do not believe . . . (2Corinthians 4:3 - 4, HBFV).
Now you were dead in trespasses and sins, in which you walked in times past according to the course of this world, according to the PRINCE OF THE POWER OF THE AIR, the spirit that is now working within the children of disobedience (Ephesians 2:1 - 2, HBFV).
Resolution
Jesus will exercise his right to rule the world when, at his second coming, he assumes its throne as "King of Kings and Lord of Lords" (Revelation 19:16).
Until Jesus' return, however, God has decided to allow the devil to continue to have authority over the world as a prince (see John 12:31, 14:31, 16:11, Ephesian 2:1 - 2, etc.). He is doing this so that all humans learn through hard experience that Satan's way of life cannot compare to the glorious existence God offers. He also allows him to exercise dominion as a means of perfecting his saints for rule during the Millennium (1Peter 1:6 - 7, Revelation 3:21, 20:4).