A Core Fragment is a piece of the Molten Core, the lair of the Firelord, Ragnaros. The fragment can be used to attune one's body to the entrance of the Molten Core and allow you to use a rift in space to move there quickly.
The quest is given by Lothos Riftwaker who is standing just outside the quarry leading to Blackrock Depths. Once you give the Core Fragment to him, you are considered "attuned" to the Molten Core. From now on, you only need to go through the window next to Lothos to enter the Core rather than venture all the way from Blackrock Depths.
If many people need to get attuned, a great way to do this is to have a party with a warlock fight their way through Blackrock Depths to the large fragment. Once everyone in the party has the fragment, remove someone from the party, invite someone who needs the fragment to the party, and then summon them once they enter the instance. Repeat this for everyone who needs the fragment.