Who is Satan?  Why is he allowed to exist? What do we know about him?

Satan was created perfect in the beginning when the angels were created. "Thou wast perfect in thy ways from the day that thou wast created, till iniquity was found in thee" (Ezekiel 28:15Thou wast perfect in thy ways from the day that thou wast created, till iniquity was found in thee.
See All...). He was "the anointed cherub that covereth" (Ezekiel 28:14Thou art the anointed cherub that covereth; and I have set thee so: thou wast upon the holy mountain of God; thou hast walked up and down in the midst of the stones of fire.
See All...), had a covering of precious stones (Ezekiel 28:13Thou hast been in Eden the garden of God; every precious stone was thy covering, the sardius, topaz, and the diamond, the beryl, the onyx, and the jasper, the sapphire, the emerald, and the carbuncle, and gold: the workmanship of thy tabrets and of thy pipes was prepared in thee in the day that thou wast created.
See All...), and had within himself the pipes and tabrets that made him capable of giving a musical concert all by himself (Ezekiel 28:13Thou hast been in Eden the garden of God; every precious stone was thy covering, the sardius, topaz, and the diamond, the beryl, the onyx, and the jasper, the sapphire, the emerald, and the carbuncle, and gold: the workmanship of thy tabrets and of thy pipes was prepared in thee in the day that thou wast created.
See All...). All of this points to a special job of praising God.

Since he is more powerful than Michael the archangel (Jude 1:9Yet Michael the archangel, when contending with the devil he disputed about the body of Moses, durst not bring against him a railing accusation, but said, The Lord rebuke thee.
See All...), he must have been the most powerful of God's creation. As Lucifer (Isaiah 14:12How art thou fallen from heaven, O Lucifer, son of the morning! how art thou cut down to the ground, which didst weaken the nations!
See All...), he was the light-bearer. Since in his rebellion he determined to exalt his throne (Isaiah 14:13For thou hast said in thine heart, I will ascend into heaven, I will exalt my throne above the stars of God: I will sit also upon the mount of the congregation, in the sides of the north:
See All...), it figures that he must have had a throne given to him by God. In Luke 4:5-6 [5] And the devil, taking him up into an high mountain, shewed unto him all the kingdoms of the world in a moment of time. [6] And the devil said unto him, All this power will I give thee, and the glory of them: for that is delivered unto me; and to whomsoever I will I give it.
See All..., Satan claimed that the kingdoms of the world had been given to him and Jesus did not dispute the fact. So, not only was Satan a special instrument of praise directed toward the Almighty, he must have been given a special position of trust and importance in the heavenly hierarchy.

His fall into sin and rebellion came because of his own pride. Timothy is warned against placing a man into the office of bishop who is a novice, "lest being lifted up with pride he fall into the condemnation of the devil" (1Timothy 3:6Not a novice, lest being lifted up with pride he fall into the condemnation of the devil.
See All...). Therefore, we see that the devil was condemned because he was lifted up with pride. This fall is spelled out in the following passage:

Isaiah 14:12-15 [12] How art thou fallen from heaven, O Lucifer, son of the morning! how art thou cut down to the ground, which didst weaken the nations! [13] For thou hast said in thine heart, I will ascend into heaven, I will exalt my throne above the stars of God: I will sit also upon the mount of the congregation, in the sides of the north: [14] I will ascend above the heights of the clouds; I will be like the most High. [15] Yet thou shalt be brought down to hell, to the sides of the pit.
See All... How art thou fallen from heaven, O Lucifer, son of the morning! how art thou cut down to the ground, which didst weaken the nations! For thou hast said in thine heart, I will ascend into heaven, I will exalt my throne above the stars of God: I will sit also upon the mount of the congregation, in the sides of the north: I will ascend above the heights of the clouds; I will be like the most High. Yet thou shalt be brought down to hell, to the sides of the pit.

Satan determined to lift his throne above the stars of God and brought condemnation on himself. Your final question about why God allows Satan to exist is not directly answered in scripture to my knowledge. Jesus spoke of "everlasting fire, prepared for the devil and his angels" (Matthew 25:41Then shall he say also unto them on the left hand, Depart from me, ye cursed, into everlasting fire, prepared for the devil and his angels:
See All...). Therefore, the final destruction of the devil is predetermined. In the end, "the devil that deceived them was cast into the lake of fire and brimstone, where the beast and the false prophet are, and shall be tormented day and night for ever and ever" (Revelation 20:10And the devil that deceived them was cast into the lake of fire and brimstone, where the beast and the false prophet are, and shall be tormented day and night for ever and ever.
See All...). But for now, he is allowed to continue as "the god of this world" (2Corinthians 4:4In whom the god of this world hath blinded the minds of them which believe not, lest the light of the glorious gospel of Christ, who is the image of God, should shine unto them.
See All...) and "the prince of the power of the air" (Ephesians 2:2Wherein in time past ye walked according to the course of this world, according to the prince of the power of the air, the spirit that now worketh in the children of disobedience:
See All...).

For the reason of the devil's present freedom, I must speculate a little. I find it instructive that though he is chained and kept in the bottomless pit during the thousand year reign of Christ on the earth (Revelation 20:1-4 [1] And I saw an angel come down from heaven, having the key of the bottomless pit and a great chain in his hand. [2] And he laid hold on the dragon, that old serpent, which is the Devil, and Satan, and bound him a thousand years, [3] And cast him into the bottomless pit, and shut him up, and set a seal upon him, that he should deceive the nations no more, till the thousand years should be fulfilled: and after that he must be loosed a little season. [4] And I saw thrones, and they sat upon them, and judgment was given unto them: and I saw the souls of them that were beheaded for the witness of Jesus, and for the word of God, and which had not worshipped the beast, neither his image, neither had received his mark upon their foreheads, or in their hands; and they lived and reigned with Christ a thousand years.
See All...), he is released for a season at the end of the thousand years so he can gather the nations to battle against the Lord (Revelation 20:7-8 [7] And when the thousand years are expired, Satan shall be loosed out of his prison, [8] And shall go out to deceive the nations which are in the four quarters of the earth, Gog, and Magog, to gather them together to battle: the number of whom is as the sand of the sea.
See All...). When this is taken with the appearance of Satan as the serpent in the Garden of Eden, it seems that God does not want us to follow Him simply because we have no other choice. He wants heaven to be populated with people who have chosen Him over the offers of the devil. That, in my opinion, is the main reason Satan is presently allowed access to men--to purify our choice to love and serve God.