There is a biblical explanation
to this long-asked question. It involves Satan's implied challenge
to God's right to rule humanity
when he tempted Eve
. He intimated that God
was unrightfully withholding something from the woman
and also implied that God was a liar
when he said that she would die if she ate the forbidden fruit
. Satan also made her believe she would be free
from God, becoming like God herself. Satan in effect raised himself higher than God in Eve's eyes, and by accepting his offer, both Adam and Eve showed that they did not want God's guidance in matters of right and wrong
Instead of responding to this challenge by beating Satan to death, God decided to give him and humanity a chance to prove that they does not need God to guide them. As we see in chapter 6 of Genesis, Satan's angels came to the earth and filled it with violence and wickedness, only to return to spirit form when the floodwaters came. Understandably, God did not allow them to occupy their former positions as his servants in heaven, but he did let them exist in the heavenly realm for several thousand years.
At the other end of the bible, in Revelation we read that Satan and his angels were eventually kicked out of heaven by Jesus and his angels. Chapter 12 and verse 12 says: "Therefore rejoice, ye heavens, and ye that dwell in them. Woe to the inhabiters of the earth and of the sea! for the devil is come down unto you, having great wrath, because he knoweth that he hath but a short time."
So basically Satan is maneuvering things, doing what he can to turn mankind away from God and prove him a liar before he is abyssed and eventually destroyed, and in the meantime God is letting him carry on and show himself up as a liar himself because, as we can see, humankind has made a bit of mess of the earth thus far.
"But why not just destroy Satan in the first place?" you may ask. God could have done this and avoided all the chaos and misery we know today, but it wouldn't really have proved anything. Anyone could rise up afterward and say "hey, you didn't give that guy a chance to prove he might be right after all," and so God would look a bit silly and nobody would like him.
In summary, God exists, and he isn't evil or powerless - he's just waiting.