To get a building material or an item necessary to complete a quest from a neighbor, press the "Ask" button. In this case a request for help will be sent to you neighbors. Only 1 neighbor can react to each request. In order for the item sent in response to your demand to appear in your inventory, you need to thank the neighbor who helped you. To do so, open Requests tab in your post box and click "Thank you" button or press "Yes" next suggestion "Thank everyone". 
Every 15 minutes you can send 1 request to your neighbors. In case you didn't open the game for 5 hours, you can send 2 requests at once.
If you have been absent from the game for 12 hours, you are able to sent 3 requests, and if you have been missing for 24 hours - 4 request are available for sending.

Take notice that items in your Mailbox will disappear after some time, so it is good to collect them as soon as possible.