This is a really deep question that requires a fundamental understanding of electrons, atoms, and the wave-like nature of light (or electromagnetic radiation in general). However, let me give you a glimpse into why this happens.

When you say mirror, let's assume we're talking about a silver metal coating, and this is also why other metals are shiny. All atoms have electrons, and these electrons can interact with light. In the case of metals, these electrons are only loosely attached to the metal atoms, so they can move around (which is related to the fact that metals conduct electricity). When light hits the metal, the electrons interact with the light and cause it to reflect. Specifically, light with a given frequency causes the electrons to rattle with that same frequency. In the case of visible and infrared light, which are low frequency, the electrons can match the speed and reflect the light. However, in the case of UV light, the frequency is too high, and the electrons can't rattle fast enough to match the frequency of the light. For this reason, metals are actually transparent to UV radiation!!! So, if you were underneath a piece of metal, it could be totally dark (because no visible light gets through) but still get sun-burnt! Neat!

Text added to this answer from the author on December 2015:

The nature of the metal will determine what wavelengths can pass through. The highest frequency light that can be scattered (that is to say, reflected) is referred to as the plasma frequency, and is different for each metal. For example, aluminum is transparent below 90 nm, silver is transparent to UV radiation below 140 nm, platinum is transparent below 240 nm, and potassium is transparent below 330 nm, which is almost getting to the visible spectrum of light! (Visible light ends at about ~400 nm.)

All this said, if you're behind a piece of steel, you're not going to get a sunburn.

Mirrors reflect mainly because they are electrically conductive. Light is an electromagnetic field, and when it hits a mirror the metal inside of it (usually aluminum or silver) cancels out the electric field parallel to the mirror which causes it to change directions and reflect away. Not all reflective mirrors are conductive though - a certain percentage of light will reflect wherever there is a change between two types of materials like air and water, because the electric field changes as it goes though different materials some of the light will get reflected while the rest will pass though or be absorbed.

Thanks for the great question! Mirrors reflect light because of how "difficult" it is for light to travel through the material that the mirror is made of. A very common mirror materials is highly-polished silver.

To understand why mirrors reflect, first think about light in outer space where there is no atmosphere. In outer space light can travel without being reflected, scattered, or impacted by any surrounding atoms or molecules (like mirror atoms or gas molecules). Once light enters the Earth's atmosphere, some of the light begins to be impacted by the gas molecules in the Earth's atmosphere, but this impact is very small so the light can still keep traveling forward very "easily".

Once the light gets to the surface of a silver mirror, the light cannot travel through the silver, but the silver also cannot absorb the light. As a result, the light "bounces off" of the surface of the silver and returns to your eye, which is why you can see yourself in a mirror.

However, it is important to keep in mind that light can travel through some solids (window glass) and be absorbed by other solids (solar cells), so lots of interesting things can happen to light at solid surfaces! It is also interesting to know that light always travels at the same speed (~671,000,000 miles per hour!) whether in outer space, the Earth's atmosphere, liquid water, or solid window glass.

Note from ScienceLine Moderator:
A reader sent the following correction to this paragraph:
"The speed of light is not constant in all materials at all. In the case of liquid water, it would be 3/4 c. That is directly related to the 4/3 value of the index of refraction of water."

I hope that this answer helps, and please do not hesitate to write back to us if you have any additional questions!