When most people use the term "mirror," they are referring to what is known as a plane mirror. This object takes the light that hits it and reflects it back. Those used for common consumer purposes are of this sort.
A mirror is essentially a highly reflective surface. The type that people see on walls or in bathrooms are known as back-silvered mirrors. This means that the reflective surface — in most modern ones, this is aluminum — is viewed through a thin layer of glass. The glass protects the aluminum from scratching and bubbling, but also distorts the image somewhat.
Early mirrors were created by simply polishing a suitable substance until it became highly reflective. Examples from the Neolithic era have been discovered, made by grinding down obsidian rocks and polishing them to an incredible sheen. They have remarkable properties, allowing even subtle details to be clearly seen in their reflections.
To make such a device, a person first needs to find the right stone: obsidian, ideally. A rougher stone is used to grind the base stone down to a flat surface on one side. After this is achieved, a finer grinding stone and clay slip can be used to polish the stone to a fully reflective sheen. Extremely fine abrasives, such as ash, can help further. After a substantial amount of time and effort, the person will have created a very primitive form of mirror.
Modern ones are made using an entirely different process. Allowing liquid metals to condense on a sheet of glass can provide a surface far more reflective than anything achieved by polish. This can be done at home with only a few supplies easily acquired at a local chemistry shop.
With pure silver nitrate, distilled water, and ammonia, a person can make a mirror virtually indistinguishable from those purchased at a store. The process (in abbreviated form) involves dissolving a small amount of silver nitrate in distilled water, then adding diluted ammonia until the mixture goes through distinct chemical changes. A second mixture is made using silver nitrate and Rochelle salts. This mixture must be boiled and filtered. By pouring these mixtures on to a very clean piece of glass sufficiently heated to the proper temperature, the silver will precipitate and form an even coating on the glass. After drying, the back of the silver can be coated with a solid paint to help prevent degradation of the silver.
Commercial mirrors are manufactured in more or less this same manner, though materials such as aluminum might be used instead of silver. Ones made for specialized purposes, such as those to be used in lasers or telescopes, are manufactured using much more exacting techniques to achieve much greater precision, but the general principles remain the same.