How do online shops work?

It can work just about anyway you want it to. Primarily though, you first sell an item on your store, collect the money, then purchase that item from the provider, at an agreed upon discount, then the provider simply ships it directly to your customer...with your company label when applicable.

There are a number of services out there that just let you plug into their system which makes many of those steps, and access to products much easier.

Online shops come in a wide variety, and the way they work can be of several generic types:

Direct Sales (You make the sale from your site, sales page, or online auction like eBay)

Generally, direct sales allow you to get access to the revenue almost immediately, so many folks prefer these methods. However, they do require additional effort on your part. In each case of direct sales, you either OWN the product or have rights to sell it from your "business".

Physical Products, locally warehoused and shipped
You sell the product, and upon notification of sale (and receipt of revenue) you ship the product from your warehouse/room/garage.
Disadvantages of this method is the need to warehouse the product, and shipping materials, and perform the physical packing and shipping processes. This means you have capital tied up in inventory and supplies, and also have labor costs for packing and shipping, plus the actual shipping costs (which you usually charge the customer for). Also, you can incur losses if you don't sell all of your inventory, it ages, is damaged or otherwise becomes unsaleable.

Advantages to this method are primarly related to having an exclusive or unique product. It is the "typical" method of most brick and mortar stores and shops, as well as the initial online shops.

Physical Products, Drop Shipped by a fulfillment house
The response by Doran Peck essentially describes this process.
You make the sale, then notify the fulfillment house to ship the product for you. This can be set up in an automated method so you don't need to even send the notification, but the automatic systems can be problematic depending on how you are getting paid.

Disadvantages are that you are selling the same old products as everyone else, and unless you have some compelling sales copy, folks are going to just buy the one they find the cheapest... Also, because you never see the product, what the customers get (and in what packaging) may be different than you and they expect. Looking at building a business, many drop shippers have inserts that send them to their shops for follow on sales...

Advantages are no need to tie up capital for inventory or supplies, nor hire folks to do your packing and shipping.

Downloadable product
You make the sale, collect the revenue (usually through a shopping cart and merchant account, or through a service like PayPal), then forward the download link to the customer. Download can be from your own server or a "cloud service" such as S3.amazonaws.com.

Disadvantages include compatibility with the download platform (PC, Phone, Tablet, etc) especially for software, or perceived value of things like ebooks related the expectations set on your sales page. (If you are selling the answer to all of their problems, and they get a 5 page ebook, there may be a disconnect on expectations.)

Advantages include "instant gratification" for customer, lack of shipping hassles, easy repeatability, etc.

Affiliate Sales

You send the prospect to the sales page of the vendor, and if they turn into a customer, the vendor takes care of shipping/fulfillment/download and you get a commission. Usually there is some delay between purchase and receiving a commission.

Disadvantages: You often don't know who actually purchased, so you can't maintain a "buyers" list. Conversions are subject to sales copy on the Vendor's site. Rarely, you may encounter a less-than-ethical vendor that doesn't give you credit for your sales. You have to make sure links are properly coded with your affiliate ID to get paid at all. Commissions are generally delayed by weeks or more, depending on the agreement, to account for chargebacks, refunds and the like.

Advantages: You send the prospects, you get paid if they buy. No product to develop, maintain, protect or download. Easy to get started. Can change your business model or expand into other niches without significant issues. Vendor takes care of all fulfillment and return actions. You can focus on finding the customers.

Rick's basically spot on.