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 28:15). He was "the anointed cherub that covereth" (Ezekiel 28:14), had a covering of precious stones (Ezekiel 28:13), and had within himself the pipes and tabrets that made him capable of giving a musical concert all by himself (Ezekiel 28:13). All of this points to a special job of praising God.
Since he is more powerful than Michael the archangel (Jude 1:9), he must have been the most powerful of God's creation. As Lucifer (Isaiah 14:12), he was the light-bearer. Since in his rebellion he determined to exalt his throne (Isaiah 14:13), it figures that he must have had a throne given to him by God. In Luke 4:5-6, 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 3:6). 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 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 25:41). 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 20:10). But for now, he is allowed to continue as "the god of this world" (2Corinthians 4:4) and "the prince of the power of the air" (Ephesians 2:2).
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), 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). 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.