Gifts are an in-game actions where neighbors can send items from storage or daily gifts to other neighbors.
To send Daily Gifts you can select the gift box towards the top left of your screen. There is no limit to the amount of daily free gifts one can receive.
Special Gifts are anything which comes from another neighbors storage such as things on your wish list. There is a limit of 55,000 coins. You can upgrade this limit to 255,000 by upgrading the barn to a granary. Another way to up the limit is to purchase emerald expansions from the market (Market > Buildings > Expansions). You can see a persons gift limit by selecting an item on their wish list and looking about the avatar. You can see your gift limit by selecting the gift box above your friends list and selecting the special tab.