There are 3 sizes of Power Stones: Fragment, Shard, and Crystal.
It is possible to obtain Power Stones and keep them before you have built the Sculptor, however those that are will only be obtainable through the Daily Reward and Supply Chests. Any Power Stones you earn are stored on your base. You will not be able to see how many Power Stones you have until the Sculptor is built, nor are they usable until then.
Any Power Stones that you own cannot be lost to attackers. If your home base is destroyed and the attacker wins one or more Power Stones, the Power Stones they get are freshly generated by the game as a part of the attacker's loot bonus.
Power Stone Chance
When a base is destroyed, there is a 25% chance that the Headquarters will give you a Power Stone. Every other building on the base has a 2% chance that it will drop a Power Stone.
If you do get a Power Stone from a building, there is an 85% chance it will be a Fragment, 12% chance it will be a Shard, and a 3% chance it will be a Crystal.
A certain type of Dark Statue boosts Power Stone chances. The extra chance percentage given by the Statue is multiplied with each building's base chance. For example, if you have a total of +100% Power Stone chance from your Statues, the enemy's Headquarters has a 50% chance of giving a Power Stone, and every other building has a 4% chance.
Certain Tribes may provide boosts that increase Power Stone chance. When used, they work in the same fashion as the Dark Statues that boost Power Stone chance.
The number of buildings that you destroy before destroying the Headquarters does not affect your Power Stone Chance in any way, as every building will explode when the HQ is destroyed and they all have the chance to contain a Power Stone.
You will always get at least one Power Stone from Dr. T levels and Boss Bases always give a predefined set of Power Stones. The above chances do apply to these levels, however. You can get more Power Stones in addition to the guaranteed ones, though this is not guaranteed.
Fragments are the smallest pieces of a Power Stone. When seven Fragments of the same type are combined, they create an Idol, the smallest and least powerful Statue.
Fragments can be automatically converted to Shards without the need to construct a single Idol; by tapping the type of Fragment to be converted, and providing confirmation, the amount of Fragments you have will be converted into as many Shards as possible, with leftover Fragments remaining in the inventory. Note: Converting fragments will not count towards the Perfectionist Achievement.
This can even be done while the Sculptor is creating a statue.
Shards are the medium-sized Power Stone. Combining seven Shards of the same type creates a Guardian, the medium-sized Statue. Guardian Statues are guaranteed to be more powerful than even the best possible Idol.
Idols may be salvaged to receive one Shard of the same type of Power Stone used to create the Idol.
Crystals are the largest Power Stone. Combining seven Crystals of a single type create a Masterpiece, the largest and most powerful Statue. Masterpiece Statues are guaranteed to be more powerful than even the best possible Guardian.
Guardians may be salvaged to receive one Crystal of the same type of Power Stone used to create the Guardian.
Power Stone Types
Life Power Stones are used to make Statues that boost production of resources. This bonus increases the output of the player's production buildings, freed NPC bases (Gold only), and Resource Bases (depending on specific Statue bonus). Life Power Stones can be received from Tropical enemy player bases, NPC-controlled bases, and some Lt. Hammerman bases.
Magma Power Stones are used to make Statues that increase troop damage and health. Magma Power Stones can be received from enemy player bases, NPC-controlled bases, and Lt. Hammerman bases that are located on Volcanic Islands.
Ice Power Stones are used to make Statues that increase building damage and health. These Power Stones can be received from enemy player bases, NPC-controlled bases, and Lt. Hammerman bases on the northern (upper) part of the archipelago that is covered with ice and snow. When attacked, these bases will have white playing fields instead of green or volcanic.
Dark Power Stones are used to make Statues that increase Gunboat energy, Power Stone chance, and resource rewards. These Power Stones are only obtainable via the Dr. T events, randomly as part of the Daily Rewards and Supply Chests, and as possible loot from Submarine dives which will be mixed with the primary stone element from the dive spot. The first Dr. T base is available upon the completion of Dr. T's level 10 Tropical Base. Upon the completion of Dr. T's level 20 base, Dr. T's Volcano Base becomes available.
The limit on the amount of each size of each type of Power Stone one player can have is 9 million, which means you never have to worry about running out of space for more Power Stones because it is virtually impossible to reach this limit.
Power Stones gained from diving typically match the region the dive spot is found in, i.e. Ice Power Stones being found in the ice region, but any Power Stone could potenially be found anywhere in the Archipelago.