I don't know whether the books metioned in Best ever book on Number Theory are beyond undergraduate/high-school-olympiad level.
Please recommend your favourite.
There are several elementary number theory books which you could use and which do not assume a level of knowledge beyond high school math. I would strongly recommend Underwood Dudley's Elementary Number Theory and Harold Stark's An Introduction to Number Theory.
They're both beautifully written and cover most of the things that are usually covered in any introductory number theory course (at a basic level of course). Some of the other books that have already been suggested are excelent.
Particularly Katie Banks' suggestion of Ireland and Rosen, although this book makes some use of the language of groups, rings and fields, so it may be more advanced for a high school student.
My favorite elementary number theory book is the one I published with Springer: http://wstein.org/ent/ This isn't a completely shameless plug, because I was just allowed to release the PDF version legally for free (available at the above URL), which the original poster might appreciate.
There is a list at http://www.imomath.com/index.php?options=mbb|knjige&p=0 . There's also the book Problems of number theory in mathematical competitions.
Davenport's The Higher Arithmetic was my first number theory book. I think its very accessible to a high school student or beginning undergraduate student. It's quite short and very quickly readable. If you find this treatment too informal, Niven and Zuckerman's an Introduction to the Theory of Numbers is a standard text that I think is a very well written undergraduate text, but this has already been mentioned.