The proof from motion says: whatever is in motion was at rest until moved by something else, and that by something else, and so on. Unless we are to get involved in an infinite regress we must accept that there must have been a First Mover that itself is unmoved. This First Mover we call God.
The proof from efficient cause argues that everything in the universe has an efficient cause that we could call its maker. Unless we are to get involved in an infinite regress, we must accept that there must have been a First Maker that itself is uncaused. This First Maker we call God.
The argument from contingency looks at the idea we might not have been. If everything were merely possible, there would have been a time when nothing yet had come to be. Nothing comes from nothing, so in that case there would be nothing now. However there is something now — the world and everything in it — so there must be at least one necessary being. This necessary being we call God.
Aquinas listed objections to his views first.
Here he gives arguments that others might have given against his objections.