Well, to answer this question, we first need to understand what light is. You may have heard of electric and magnetic fields: electric fields are created by charged particles, and magnetic fields are created by moving charges. It turns out that electric and magnetic fields can create each other: if an electric field changes in time, it creates a magnetic field, and if a magnetic field changes in time, it creates an electric field. So electric and magnetic fields are really the same thing, and we call them the electromagnetic field.

Now, let's say you have an electric charge (like an electron, say) and you start wiggling it. As you move this charge around, you create an electromagnetic field that changes in time; the magnetic and electric fields create each other, and you get oscillations (or wiggles) in the electromagnetic field that move outwards and away from the charge. These oscillations are called electromagnetic radiation. It turns out that if you wiggle the charge just right, the electromagnetic radiation you get is just the light we see! So light is just waves in the electromagnetic field.

Now, what happens when light hits something? Well, matter is made out of atoms, and atoms contain electrons. When electromagnetic waves hit these electrons in matter, the electrons start jiggling around, since they're charged. But as they jiggle, they create more electromagnetic waves that travel outwards and away; these electromagnetic waves are the reflected waves. Depending on how the atoms and electrons are distributed, these reflected waves can either be all lined up, or go off in random directions. If the reflected waves go off in random directions, we call that diffuse reflection; this is what you see most of the time, when light hits an object and illuminates it, but you can't see a reflection in it. However, when light hits an object and is reflected all in the same direction, we call that specular reflection; this is what happens in mirrors, where you can see your reflection.

White light reflects off of all surfaces to some degree, but what makes mirrors special is that they are extremely smooth so that very little light is scattered off of rough parts all directions. This means nearly all of the light comes right back at you. Essentially, you can think of rough parts of a material being made up of tiny, smooth surfaces that angle in different directions. When light hits these angled surfaces it reflects off in a different direction and you 'lose' that light. When a surface is mostly smooth, most of the light is reflected right back at you and you see an undistorted image.

If you question is a more fundamental one, that is why does light reflect off things at all, then the answer is that light interacts with the electrons in materials. When a photon 'collides' with a material surface, it is absorbed by the electron cloud which promptly relaxes in any number of different ways but one of which is simple re-emission in a different direction. Doing out the math you find that the lion's share of probability lies in the photon being re-emitted at the same angle it initially impinged on the surface.