There are 2 ways to send a gift to a neighbor.
1) From the inbox
Go into your inbox, the yellow envelope at the top right of your screen.
Tap on the "Gifts" tab and then move down to the "Neighbors Wishes" section. Tap on send to give someone the gift or the X to ignore the request.
After you have cleared all your neighbor's gifts and requests, you can send out free gifts.
2) From the MENU
Tap the menu button, then the Gift icon and then Free Gifts.
You can send an item to a neighbor by selecting the "Send" button. More items will be added here when you reach higher levels.
When you have tapped on the "Send" button, the neighbor panel shown below will be displayed. Tap on the green plus to move the player over to the selected list. Players who appear in the selected list will be sent the item selected when you tap the green "Send" button. You can use the search bar at the top to find a particular neighbor or use the "Select all" button to send the selected item to all your neighbors. Tap the green "Send" button to send the gift your neighbors.
After you have tapped on this button, a sent successfully message will appear at the top of the panel.
Note: You can only send gifts once every 24 hours to each individual neighbor. The 24 hour limit starts from the moment a gift is sent to a neighbor.