Refined Storage is a mod that allows you to store a massive amount of items digitally. That means instead of using Chests and Drawers, you use Disks and Grids.

To start, you have to place the Controller in the world. Every other part of the system must be connected to the Controller using Cables. The Controller is also where the RF that powers the system is provided.

Connect a Disk Drive to the Controller and place some Storage Disks inside. You can have up to 10 Storage Disks of any kind in a Disk Drive.
Disks store items as if they were files in a computer. Disks with more storage space can store more. A 1k Storage Disk can store 1,000 items, 4k can store 4,000, and so on.

The Grid allows you to access items in the Storage Disks. Item stacks have no limit, so if you have more than 64 of an item it'll stay in one slot.
You can search for items in the text box at the top. 

The Crafting Grid has a 3x3 crafting table built in, which can use items stored in the system.

You can move fluids in and out of the Fluid Grid by placing Buckets of that fluid into it like any other item, which will empty the Bucket. To take fluids out, click on the fluid and it will try to fill a Bucket from either your inventory or the storage network.

Fluid Storage Disks store fluids instead of items. A 64k Fluid Storage Disk can store 64,000mB (64 buckets) of fluid, etc.