A Philosopher's Stone is an immensely powerful object that amplifies all uses of Alchemy. This stone, despite how powerful it is, only has a limited amount of power and can only be used a certain number of times, due to the the Law of Equivalent Exchange. A good portion of the story line in the original series, Brotherhood series, and the manga, revolves around Ed and Al trying to get a Philosopher's Stone in order to get their bodies back.
The Philosopher's stone is red in color with a purple tint to it. It can come in many different forms, most notably a rocky coal-like form (most commonly seen in the anime), a round stone (as seen in multiple Homunculi's bodies), or it can come as a liquid. The smaller a Philosopher's Stone is, the less power it contains.
The exact process to make a Philosopher's Stone is unknown, however it is known that that the main ingredient in Philosopher's Stones is the life of humans. The more souls concealed in the stone, the more powerful it will be. The Philosopher's Stone uses the the lives of humans to override the Law of Equivalent Exchange and destroys the souls of humans every time it is used. When all of the souls in the stone are used up, the stone itself then disappears.