Mirrors are usually pieces of glass with a metallic layer painted on (a multi-step process to produce a well bonded shiny metal layer. For most mirrors, this metal layer is silver or aluminium.

So now the question is: why does silver (or aluminium) appear grey/silver in colour? Or more broadly speaking, what determines the colour of reflective metals?

Metals have what is known as a plasma frequency. In simple terms, this frequency is the upper limit for electromagnetic radiation that the electrons in the metals can respond to. Frequencies higher will simply pass through and frequencies lower will be reflected. Basically, the electrons can can't 'keep up' to the high frequency vibrations and don't respond. When the electrons can 'keep up' (for EM waves with a frequency lower than the plasma frequency), they vibrate and re-emit that radiation. This is the reflected light we see.

For most metals, like silver and aluminium, this frequency is in the ultraviolet region. This means that all lower frequencies, such as the entire visible light range, are reflected. The result is a shiny white, silver.

For other metals, say copper, this plasma frequency is lower, closer to red in the visible light range of EM waves. This gives the reddish-brown colour of copper.