There are some passages that make it sound that the Holy Spirit is Christ's Spirit, but according to our Trinitarian Theology, each person of the Trinity is unique and also God. That means, The Father is not the Son, the Son is not the Spirit, and the Spirit is not the Father. Yet, all three are God. They act independently of each other and play their own roles.
HOW DO WE TALK ABOUT IT?
The first step to understanding the Holy Spirit is to not be afraid to talk about him (or her, whatever you prefer). You can ask the Holy Spirit to give you strength when you feel you are being temped. We should also talk about this with our children. Just because we adults have a hard time understanding the Trinity doesn't mean we should keep it from our kids. The sooner they start to hear about it, the sooner they will grasp it.
The Holy Spirit is the way God gives us power over sin in our life. We can use that power to live a holy life according to God's will. That's something children can understand.