You are not searching the entire web when you search a search engine. Instead, you are searching that search engine's database of web sites. All search engines are not created equal, they operate differently, and they contain unique databases of unequal size and currency. The best strategy is to get familiar with a few, and try your searches with more than one.
Most search engines have both basic and advanced search modes.
You will probably be able to further refine and qualify your search request if you use the advanced mode. To help you get the most out of a particular search engine, you should check out the Help link that is usually available from the search engine home page. Try some searches in a couple different search engines, using both the basic and advanced search modes. You are now in the Basic Search Option. Now lets try and narrow the search by looking only for information from non-profit sites.