If you die in a dungeon, you can spend 1 Magic Stone to revive with full HP. This is not recommended for Non-IAP players.
For 1 Magic Stone, you can restore your stamina to full. This is also not recommended for Non-IAP players.
Expand Monster Boxes
You can spend 1 Magic Stone for 5 additional Monster Boxes, which lets you store 5 more monsters. This is essential as you progress in the game.
Once you've reached 50 friend slots from ranking up, you will unlock the option to spend 1 Magic Stone for 5 additional Friends List slots.
Rare Egg Machine
You can spend 5 Magic Stones on one roll for a chance to receive various rare monsters. For more details, see Egg Machines.
How to Get Magic Stones
Clearing a dungeon for the first time
Clearing all levels of any dungeon (normal or limited-time) will give you a Magic Stone.
A limited-time dungeon's "Clear" status does not reset after it closes and re-opens, so you can only ever get one stone from it.
The only exception is if there were new levels added to the event dungeon after you cleared it, because this is technically a different dungeon in the game data. Beating the new levels will give you a second Magic Stone.
Cumulative log-in bonus. For more details, see Log-in Bonus.
Official events. Happens about twice a month.
Compensation from GungHo for bugs.
And of course, spending $$$...
Advice for Non-IAP Players
In the early game, since Magic Stones come so easily through clearing dungeons, it's fine to roll the Rare Egg Machine a few times to get strong monsters for your team. It's highly recommended to do this during Godfests only, otherwise there's a high chance of getting something less useful. There is one Godfest per event, so likely every 2-3 weeks. The "Rare Egg Machine" button will have yellow text saying "Godfest Active".
Once you have a solid team, spend Magic Stones as needed to expand your Monster Boxes to store evolution and feeder materials.
Outside of very specific circumstances, do not use Magic Stones to revive, because if a dungeon kills you once, it will probably kill you again. If you need to ask, you don't need to revive.
Think of it this way: 5 stones = 1 pull at the Rare Egg Machine. If you spend x stones to beat a dungeon, will you come out with a monster that you feel was worth x/5 pulls?
You can spend stones to recover stamina once your rank is high enough that it's not practical to wait or rank up to refill it. Non-IAP players are advised to only do this for certain urgent Dungeons such as Super Metal Dragons Descended, and only if they have enough stamina to make it worthwhile (e.g. 100 stamina for 2 runs at a 50-sta dungeon).
Buying Magic Stones
Magic Stone price list (Rates current as of July 2, 2013)