Basically, if you're trying to build an oven or have a goal that specifies you request items you can ask your neighbors for them. It has to be someone that's on your neighbor list, so you'll want to have at least 20 active neighbors at all times since that is the limit of requests and gifts you may receive per day.
Requesting oven parts: Once you've purchased and placed the oven, it will appear as a box full of junk. Double tap the oven and you will get a pop up of what parts you need to build it. If you don't have enough parts there will be buttons beneath the part's photo that read "Ask Friends" or "Buy for 2 Gems". Simply tap the "Ask Friends" button and your Neighbor List will appear, select each neighbor you'd like to ask and press "Send." The next time your neighbors open their apps, your request will appear in their News Feed. Now all you have to do is wait to get enough parts to build the oven.
*Note: if you choose to click "Buy for 2 Gems" instead of waiting for the requests to come through, you will only be purchasing a single part, not the total amount needed to complete the oven. For example: if you need 10 Screws, you would have to click the "Buy" button 10 times, each time paying 2 gems to get one screw.*
If you're trying to request items to complete a Goal: Tap on your Goal Book, then tap on the specific goal, then click the button that says "Ask" and the rest follows the same as requesting parts. You'll get your neighbor list, select the neighbors you want to ask and click "Send".
You can only get 20 gifts and 20 requests per day so it's important that you're sure your neighbors are active so you don't waste time waiting for items from a player who has quit the game. There's a thread that outlines all the important things players should know, it's called the Bakery Story Game Guide, I'd recommend giving it a once over so you're not surprised by anything that pops up in the future.
https://forums.storm8.com/showthread.php?59840