'Spiritual warfare' is the struggle to have life in this material world reflect as much as possible God's loving governance. It is a 'war' because there are forces working vigorously to thwart what God is doing. God is in charge, but there is an enemy that is in full-scale revolt, and it has powerful influence all around. As with the unseen God, the forces behind the revolt are unseen, non-physical, and supernatural. They lust after power in the world of visible, material beings. Just because the battle is unseen doesn't mean it isn't going on. It is. In every nook and cranny of our earthly existence. In deciding to follow Christ, the believer accepts the rulership of Christ in his/her life (that's what's meant when Jesus is called 'Lord' -- His authority and rule). This New You yields the throne of the self to Jesus, but the Old You doesn't like it one bit.
The struggle against the Devil and all his empty promises is at its heart a 'second Person' matter, a work of Christ Himself. The Holy Spirit leads us in our part of the struggle.
The Scriptures speak of spiritual warfare in several places, but most directly by Paul in Ephesians 6, where he speaks of the full armor of God. Most pointed is verse 12 : "For we struggle not against flesh and blood, but against the kingdoms, against the powers, against the world leaders of this darkness, against spiritual wickedness in the high places".
The Devil has already lost. But the Devil is trying to take as many with him as he can, apparently just to hurt God. And however much the Devil succeeds in doing this, God hurts -- like a parent does over losing a child. This sort of spite is very much in keeping with the Devil's character. Satan is a leech; he is so lacking in life that the only life he has left is what he can suck out of the living.