Layers are Photoshop's way of organizing an image. In a Photoshop image, each layer is analogous to a sheet of trace paper in a traditional overlay (see figure below): each is its own composition, entirely independent of all other layers, and can be edited, repositioned, or deleted without affecting the others.

In all honesty, you can accomplish just about anything without layers that you can accomplish with them. But listen up: layers make it easier to make changes, and harder to make mistakes. So it's a good idea for you to become very comfortable learning to use and manage them.