Google search provides website owners with the chance to offer a search tool on their site, the results of which contain paid Adwords advertisements, which in turn generate revenue for both the publisher and of course, for Google.
The reason why this tool is effective for publishers is this. A visitor has been searching for something which makes him end up on your site. Unfortunately, you haven't got the information he wants so he needs to continue searching and if you haven't got a search box, he'll almost certainly return to Google to carry on with his research. However, if you have your own search engine on your site returning results from Google, he will probably use that and also very probably click one of the links which comes up - that's how to make money with Google search.
Publishers don't usually feel that rival AdSense ads on their site are a problem because if they cause a visitor to click away they'll still make money and the same goes for search. Another advantage to the website owner is that the search box doesn't use up much real estate on their site and the settings can dictate that the search results appear on a different page - either a Google page or their own page in frames. Even if you don't choose for the search results to appear on your own pages, you can specify the colour scheme for the Google results page so it still looks as though it's part of your site. As these results are in a new page, your own page will remain open for you visitor to return to.
Your website will have more appeal to visitors if you offer that extra facility of a Google powered search; your visitors may even choose it as their home page if they like your content enough and find what you offer to be useful.
Many Google search publishers say that search often doesn't make as much money as AdSense units do but it can keep your visitor on your site while still earning you income for each other site that they click on. Your reader can also use Google search to search your own website so it's really multi purpose.
Adding a Google search box to your site is very simple and takes barely any time at all if you follow the simple instructions in Liz Canham's post How to Add Google Search to Your Site which you can find on her website AdSense Tips and Tricks.