Reflection occurs when light bounces off of something. That's really all there is to it. If you shine a light on something, some of the light is reflected back towards your eyes - this is what you see. Sometimes, some of the light is absorbed by the object, and only part of it is reflected back. This is how we get things with different colors. For example, if you shine light on grass, only the green light is reflected - the rest is absorbed. For a black object, no light is reflected.
These objects usually have rough surfaces that reflect the light back in all different directions, so it gets all jumbled up. If the object has a very smooth surface and doesn't absorb any of the light (like a mirror), all of the light will be reflected back in exactly the same pattern that it came in with. This is why you see a clear image. First, light is reflected off of you. Then it's reflected off of the mirror and comes back to you in the same arrangement.
One of the most important things to understand about how light is reflected is that it will bounce off of a smooth surface (like a mirror) at exatly the same angle that it hit it at. So if you pointed a laser beam at a mirror at 25 degrees, it would bounce off again at 25 degrees:
(from The Physics Classroom
(republished on 07/28/06)