• AND finds only documents containing both terms.

AND requires every search term joined by the expression to be present.  It is more restrictive (than OR) and is a great tool for successive narrowing of searches.
 

For instance, suppose I would like to find all federal cases discussing:  "condemnation" and "taking" –  6,200 hits

• OR finds documents with either term (including documents with both).

OR requires any term connected by the expression to be present.  It is the most inclusive connector.  It will give you the greatest amount of search hits. 

This connector is good for synonyms or related topics.

 "condemnation" or "taking" – over 10,000 hits

• NOT finds documents which do not contain the term.

NOT requires that a term not be present.  It is a limiter that should be used very sparingly.  While excluding irrelevant hits, one runs the risk of excluding relevant documents.

"condemnation" and "taking" and not "crime" – 5,400 hits

Westlaw uses "but not" and Lexis uses "and not."