When you speak to Tom Nook, he gives you the option of ordering something from your catalog. The catalog allows you to order an item that you have owned before. When you order an item, it is delivered by mail.
All you need to do to add a new item to your catalog is to have the item in your pocket. Any item that you have carried in your pocket at one time or another will be added to your catalog. Please note that paintings will only get added to your catalog if they are not forgeries.
To order something from your catalog, it's as simple as going to Tom Nook, talking to him, and asking to use the catalog. After some explanation, Tom Nook will show you the catalog. It is separated by item type, so furniture is in its own section, as is clothing, wallpaper, and so on. Scroll through the lists until you find what you want. Tap the name of the item, and you will see a preview of it in the upper screen. If the item doesn't say "not selling," you can click the shopping cart button on the right to order it. Nook will ask again to make sure that you really want to order it, and if you say yes, then you will receive the item the next time the mail is delivered, either 9AM or 5PM, whichever is sooner.
Yes, you can speed up your order. Instead of having to wait a while for your items to arrive, you can just get some stationery, write some letters to your neighbors, and eventually, as your villagers send their replies, the post office will have so much mail for you that they have to deliver it before the standard delivery time. Just keep mailing letters until your mailbox has the flashing thing, indicating that you have some new mail. The letter or letters from Tom Nook will probably be there.
Sometimes Tom Nook says that he is backed up with orders. This happens when your mailbox is full. If your mailbox in front of your house has mail in it, with no room for new mail, Tom Nook will say that you can't place any more orders from the catalog. Just go to the mailbox and remove the letters to make more room. If Tom Nook still says that you can't order, try sending letters to your neighbors as described above, to force the previously ordered catalog items to arrive in your mailbox.
There are some items that say "not selling" instead of the price when you click on them in the catalog. This usually occurs with rare items. For example, the snowman furniture is not available from the catalog. The snowmen are the only ones who can give you the snowman furniture. There is not a specific reason given for the omission of these items from your catalog, but it is probably to make sure those items stay special and rare.