If you are talking about already established search engines such as Google - they simply have other profitable products and services part of which is Google AdWords which they earn from with their search engine. If you are thinking of starting your own search engine, or how newer search engines make money - it's more or less advertising i.e. there will be sponsored result which are either by Google AdSense etc or paid for by their users. Some may even include affiliate links.
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yes it is possible to earn money from search engine site.there are many programs are available on online for getting income. google adsense is the best way to get nice income.