In the new Saints Row, Collectibles are objects that are scattered all across the map. By taking a picture of one with your in-game phone, you unlock it as a decoration which can be placed inside of your headquarters.
Collectibles become available after completing the main story mission 'Office Decor'. You'll also unlock the Collectibles App on your in-game phone, which you can use to track all of the Collectibles that you've found so far.