A Normal is the direction that points outwards and perpendicular to a plane.
A Vertex Normal is usually the average normal of all faces it belongs to.
Here is a picture with vertex normals being shown in green.
A Face Normal is what people generally refer to indirectly when they refer to normals.
An easy way to visualize a face normal is imagine putting a cube on the plane/face you want the normal of. Once the cube is flat on the surface of the face, imagine a line starting from the center of the bottom of the cube touching the face and going through the top of this cube. That is the face normal.
You can bend normals to achieve complex lighting effects cheaply using a normal map. This page is a pretty good reference for all things normal maps.