AND tells the computer to return only results with both of the terms entered. 

For example, you might search for:

Cats AND Dogs

This would only show you articles or books that have both words.

It would not show you results that have only one of the terms.

AND makes your results set smaller, but more specific.

OR tells the computer or database that you want results with either of a pair of terms. It does not require that both terms are present. 

A good way to use OR is with synonyms or words with similar enough meanings that they are interchangeable.

Example:

Gasoline OR Petrol

cost OR price

OR makes your results set larger.

NOT allows you to exclude a term that is irrelevant to your search. NOT is probably the Boolean term that is used the least often.

Example:

Saturn NOT Planet (want Saturn cars, not astronomy)

NOT can be tricky; it makes your results list smaller and more specific by eliminating what you don't want. If you repeatedly retrieve unrelated articles, you might look for a term that you can use NOT with to exclude them from later searches.

Once you have a good command of simple Boolean connections, you can use them together to create efficient and powerful search strings. Some databases will have forms that let you build those relationships; in others you will have to use () to clarify your intent.  Here are some examples:

(gas OR petrol) AND economy

("Climate Change" OR "Greenhouse effect) AND consequences