I think you are referring to a website that doesn't sell stuff (or at least appears to do so). This would be a website that offers "free services". Well newsflash: Unless you are talking about a purely altruistic website, there is nothing really free. As the popular saying goes, "If you are not paying, you are not the customer, but you are the product".
Most 'free websites' make money in the following ways:
1. Advertising model - Know as much as possible about the user and bring targeted ads.
2. Freemium model - sell a free product and plan to convert some of them to a paid plan.
3. Limited period promotion - Start with the free product for a promotional initial period and plan to charge it later. For instance, 37 Signals provides free 30 day trial offer for most products and then charge if you use later. This is a tough thing to master these days.
4. Sponsorship model - If your service indirectly helps the government and/or major organizations you could ask them to sponsor your service.
5.Wikipedia model - You could get donations from your users. Many wordpress plugins, open source tools and Wikipedia do this. This could be the future of newspapers.
6. Gillette model - Printers and razors are sold less than cost, as they plan to make high margin from selling a complementary product (cartridge/blades). The printer or blade you purchased will turn worthless if you don't buy the super-high margin complementary products from the manufacturer. On the web, for instance, you could create a cloud based spreadhseet/wordprocessr that is free to edit/create documents, but charged money for exporting it as a file to the local machine. Or you could charge high for the iPhone app that can access the data natively.
7. Open Source Model - Sell the product for free and plan to make money on support, customization and installation. Most open source software follow this model.
8. Usage charge model - This is related to the freemium model. Give the product free for low usage, but charge when the user is exceeding the free limits (many storage applications such as Dropbox fall under this).
9. Zynga model - Sell products through in-app purchases or to get forward in the game.
10. Credit card model - In this model, you make your product free for one side (consumers) and use the network effects to make the other side (merchants) pay. Facebook, Yelp and other online marketplaces are now getting on to the model.
11. Upsell/Cross-sell - Sell a free product & use that to promote a premium product in the same segment. For instance, If you run a finance website, you could give stock quotes free and sell premium analyst reports and financial planning tools.
12. Build a brand - Use the free service to get brownie points/good press and use the brand image to sell premium products (directly related or not) later.
13. Affiliate marketing - Signup for affiliate programs related to your service and convert your users to customers of your affiliates.
14. Sell it to Google - Build a big user base that might attract a big buyer such as Microsoft or Google, who might use the user base to sell their premium products/services.
15. Make your next website a success - If none of the previous stuff works, you could run a free website to build your personal brand/get popular and hope to get funding for your next website.