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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.

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Depends. Hardy and Wright (at least, the portion I've read) or Niven and Zuckerman can definitely be read by a student only familiar with high school algebra. Are you asking for books that focus on math competitions? – Akhil Mathew Aug 7 '10 at 0:57
following Akhil, if you're asking for "number theory for contest math" type material, the Art of Problem Solving books are superb. For number theory as a mathematical subject, Akhil's right that both Niven and Hardy and Wright can be read with basic background. I'd add much of Ireland and Rosen to that. – Jamie Banks Aug 7 '10 at 1:44
@Akhil I would like to read those introduce Number Theory as a branch of mathematics rather than as contest material. Thanks you and Katie. – Covi Aug 7 '10 at 3:45
@Covi: Then I'm not sure how this differs from the other question. – Akhil Mathew Aug 7 '10 at 4:01
possible duplicate of Best ever book on Number Theory – Akhil Mathew Aug 7 '10 at 4:01

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.

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One curious thing about Dudley's book (maybe only in the 2nd edition?) is an entry in the index: Sex in Number Theory. There really is a reference there to another page in the book. – KCd Aug 17 '10 at 21:28

My favorite elementary number theory book is the one I published with Springer: 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.

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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.

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