Copper compiles a list of Bill of Materials containing all parts included in your PCB.
Single or grouped
By selecting "Single" you can see each individual part. Typically more important is the grouped list. In this list all items that share the same part number or package and value (see discussion below) are grouped together. You can quickly see the quantity of small parts like decoupling capacitors or resistors.
Gathering part numbers
Eagle does not have a dedicated field for setting part numbers. Most components are available from various manufacturers and configurations. There are thousands of different products for a simple 0.1uF capacitor. They have different materials, tolerances and many more parameters. When designing a PCB you typically don't care about the very specific product, that is part of the part sourcing done later when it comes to ordering and assembling a PCB.
However, many add attributes to their Eagle libraries so they don't have to part source any small component over and over again. An Eagle attribute has a name and an assigned value. You can tell Copper in which attributes it should search for part numbers. You can do that in the Settings App of iOS.
Setting the attribute name for part number and manufacturer
Go to iOS Settings App and scroll down the list until you see "Copper" in the list besides various other apps you have installed.
Export Bill of Materials as CSV
You can quickly export the Bill of Materials list as a CSV file that you can upload to Mouser or Digikey any many other distributors to fill the shopping basket automatically. After you tapped the share icon, you will be presented with the standard iOS sharing screen. You can send the CSV file per Mail or iMessage or store it in the iCloud Drive or your Dropbox. If you have Numbers installed you can even directly export it to Numbers to edit the list even further.