Puzzle pieces allow you to acquire items without purchasing them for Coins or Diamonds. You'll get them through treasure chests, level-up gifts from friends, from the monthly calendar, or, occasionally, as quest rewards. You can get the item pictured on a puzzle piece by either collecting enough pieces, or by buying missing pieces with tickets - a currency that you can get through the monthly calendar, or by exchanging unwanted pieces for them.
If you click on the puzzle piece in the main menu bar, you'll see all of the pieces you've collected, and how many tickets you have. You can trade in puzzle pieces you don't want for tickets ("Scrap"), and complete items that you do want with the tickets :)
To use tickets, simply pick a card you would like to get the remaining pieces for you in puzzle piece inventory (the puzzle piece in the main menu bar) and select the green "Complete" button.
This will either tell you how many tickets it will cost to complete and let you hit that button, or tell you you don't have enough tickets to complete it. In the latter case, it will let offer you to complete it for Diamonds instead.