Mega Stones can be used after a certain point in the storyline. Certain Pokémon have a Mega Stone attributed to it, and this Pokémon must hold it for Mega Evolution to occur. In battle, you can enable Mega Evolution from the attack menu, then select an attack.
Each Mega Evolution only lasts for the duration of the battle, and only one Pokémon per team can Mega Evolve during the battle. The Mega Evolution can result in a number of things, including changed stats, abilities, and even types.
As a note, some Mega Stones are only found in some versions of the game. If you desire another, you are allowed to trade the item. However, it must be held by a Pokémon during the trade.
Initially, you can receive Mega Stones from certain people across Kalos, but that’s only half of the story.
After the game is complete, you are able to upgrade the Mega Ring. From 8:00 PM to 9:00 PM each day thereafter, according to your 3DS clock (which will read 20:00 to 21:00), you can find other Mega Stones dotted around Kalos besides those you’re given - these will appear as a shiny sparkle on the floor.
If a Mega Stone doesn’t have “8 to 9 PM” listed, it means you get it from a person. Otherwise, it’s found somewhere on the floor from 8 to 9 PM.