Inner Healing is a broad term that covers many forms of the healing of the inner man.
As we grow and learn, from infancy, we receive information into our minds, constantly. We judge whether each piece is truth, and if we judge that it is truth, we let it become a deposit in our information bank, which is like a "hard-drive" on a computer. Because, as little children, we have very little real ability to judge whether something is true or not, we let in a lot of information that really isn't true. We continue to accept erroneous information even on into our adult lives, like the modern worldly proverb, "Look out for ol' number one." This instructs us to keep up strong walls for our protection, to trust very few, if any, people, and to spend our energy focused on our own gain and advancement, instead of considering others. This is not God's way for us to live. Many Christians' lives are being shaped by the erroneous beliefs that they hold, like in the situation just described, without them even realizing it. God tells us in His Word, in Romans 12:2, "Do not be conformed to this world (or its ungodly belief systems...my words here), but be transformed by the renewing of your mind, that you may prove what the will of God is, that which is good and acceptable and perfect."
The word "renew", in this place, doesn't mean simply "to refresh" our minds, but to "change what we believe, from worldly beliefs to Godly beliefs". The Holy Spirit is at work during our entire Christian lives, showing us the places where this renewing is needed, if we will llsten to Him. When our minds are riddled with ungodly beliefs, it is as though our minds are wounded, because they can't function like the mind of Christ, which is God's goal for us (see I Corinthians 2:16). Thus, the renewing of our minds is a healing of our minds, so that they can function in Christ-like ways, can lead us in the path that God has for us, can bring blessing to others and glorify God.
In the same way that the Holy Spirit brings a renewing/healing to our minds, He wants to renew our entire inner man, or soul (our soul is composed of our mind, will and emotions). He is at work, showing us the ways in which we have real difficulties in life (particularly with people), and He wants us to seek Him for His truth about our inner problems that create our life problems. Jesus told us in John 8:32, "you shall know the truth, and the truth shall make you free." God had Paul write for all of us, in II Corinthians 4:16, "Therefore we do not lose heart, but though our outer man is decaying, yet our inner man is being renewed day by day." God is at work on our inner man, but many simply do not want to face their inner problems, or wounds, because they are afraid of what they might feel while dealing with them. Many believe that it is best to simply "move on", and try to "overcome through their own means." However, time does not heal wounds, as the world would like to think. Only Jesus can heal these wounds.
When the grade school boy was required to speak in class, he made a glaring mistake and all the other children laughed at him. He was crushed, and wounded in his heart. This hurt him very much. Now, as an adult, whenever events in his work indicate that it is going to be necessary to get up in front of a crowd and speak, he finds someone else to do it. He says, "I'm just not a public speaker". When he considers speaking like that, he feels fear rising up in him, fear that he will experience that same old hurtful feeling of rejection. His heart, his soul, needs healing. Jesus wants to heal him through the ministry of the Holy Spirit.
The instances of wounding that I have mentioned here are relatively mild, compared to some that I've seen the Lord heal. There is no wound too great for Jesus to heal.