The main purpose of painstakingly looking for the Zygarde Cells and Cores is to reconstruct the Legendary Pokemon Zygarde. You will be able to do this once you reach the Aether Base located in Route 16 of Ula'ula Island.
Inside the base, you'll find a state-of-the-art Reassembly Unit that can interact with your Zygarde Cube.
This option allows you to create or improve a Zygarde using the Cells and Cores you've collected in the Zygarde Cube.
From here, you can choose two forms of assembly: "Using Zygarde and the Zygarde Cube" or "Using the Zygarde Cube only".
If this is your first time, select the second option, "Using the Zygarde Cube only". This allows you to create a brand new Zygarde using the available Cells and Cores in the Zygarde Cube.
If you have at least 10 Cells and Cores, you can create Zygarde 10% Forme , the dog-like and weakest form of Zygarde. Afterwards, 10 Cells and Cores will be subtracted from the Zygarde Cube.
If you have at least 50 Cells and Cores, instead of Zygarde 10% Forme, you can create Zygarde 50% Forme , which is the serpent-like Zygarde found in Pokemon X and Y. Afterwards, 50 Cells and Cores will be subtracted from the Zygarde Cube.
After you've collected a substantial number of Cells and Cores, you can pick the first option, "Using Zygarde and the Zygarde Cube", to add more Cells and Cores to an existing Zygarde (one you've already created), to upgrade it into stronger forms.
So a Zygarde 10% Forme will upgrade to a Zygarde 50% Forme after 40 more Cells and Cores are added to it.
Finally, a Zygarde 50% Forme will upgrade to its unified form after the remaining 50 Cells and Cores are added to it. This Zygarde gains the Power Construct Ability and can shift between 10% and 50% forms.
Power of Zygarde
Even after creating a Zygarde using all 100 Cells and Cores, you won't be able to witness its true Complete Forme straight away. Instead, when its HP drops below half, its Power Construct Ability will trigger, transforming it into its Complete Forme.
Be aware that a unified Zygarde can not be separated, so make sure it has the correct attributes (especially Nature) before proceeding with the operation.
Of course, you can also make the unified Zygarde right off the bat if you have 100 Cores and Cells in the Zygarde Cube.
By the way, you can create multiple Zygardes using the Zygarde Cube, so long as there are remaining Cells and Cores. For example, if you have 60 Cells and Cores, you can make a Zygarde 50% Forme and a Zygarde 10% Forme.
Trading Zygarde Cells
It might not be immediately obvious, but you can actually obtain Zygarde Cells without finding them in Alola yourself.
If you receive a Zygarde 10% Forme or Zygarde 50% Forme via a trade, you can separate them into their corresponding cells, just like for your own Zygarde(s). In fact, if you're resourceful, you can even assemble a unified Zygarde without finding any cells yourself!
A good way to obtain Zygardes from other players with little effort is by depositing a rare or version-exclusive Pokemon in the GTS.
This option lets you destroy a Zygarde you've created and return its Cells and Cores to your Zygarde Cube.
Doing so will erase everything about this Zygarde--experience points, EVs, moves, memories... everything! In essence, you are releasing the Zygarde. So only choose this option if you're absolutely certain, for example if you want a new Nature for Zygarde.
With each Zygarde Core you recover, you can use the Zygarde Cube (from your bag) to teach a special move to any Zygarde in your party. This includes Zygarde's signature moves Thousand Waves, Thousand Arrows and Core Enforcer, plus some others.
Once Zygarde has Power Construct, you can also use the Zygarde Cube to change it from 50% Forme to 10% Forme and vice versa. 50% Forme is overall stronger, but has less Speed compared to the 10% Forme. Plus it's useful for those who prefer the look of the Z-dog.