Similar to Set Lists, Part Lists are just a way to group your loose LEGO Parts to make it easier to manage.
The loose parts in your Part Lists are those that don't belong to full sets. Perhaps you bought them separately or just found a pile of LEGO lying around and don't know what set they belong to.
There are two ways to add Parts to your Part Lists - one at a time after you have found the parts via search or an inventory; or by importing a list of parts from some external file or website.
The easiest way to add Parts to your collection is if you have a file or perhaps a record of your orders from sites such as BrickLink or BrickOwl.
On the Part List page that you want to import into, click the Import/Delete Parts button. Choose a file or external site to import from. You can manually create your own parts file if you like, the easiest is probably a CSV formatted file with appropriate headings. Rebrickable will automatically detect the columns as long as you use the correct headings: Part, Color, Quantity.
Adding Parts Manually
On any page that displays Parts - e.g. search results, Set/MOC inventories, etc - you can click the Part image and a popup will appear. The popup contains some summary information as well as allow certain actions to be taken. On the 'Add this Part to List/Inventory' tab select the Part List you want to add to and click the 'Add Part' button.