Books on Number Theory for Layman Books on Number Theory for anyone who loves Mathematics?
(Beginner to Advanced & just for someone who has a basic grasp of math)
 A: Number Theory For Beginners by Andre Weil is the slickest,most concise yet best written introduction to number theory I've ever seen-it's withstood the test of time very well. For math students that have never learned number theory and want to learn it quickly and actively, this is still your best choice.
For more advanced readers with a good undergraduate background in classical analysis, Melvyn Nathason's Elementary Number Theory is outstanding and very underrated. It's very well written and probably the most comprehensive introductory textbook on the subject I know,ranging from the basics of the integers through analytic number theory and concluding with a short introduction to additive number theory, a terrific and very active current area of research the author has been very involved in.I heartily recommend it.
A: Manin and Panchishkin's Introduction to Modern Number Theory
A: One book I think everyone should see is the one by Joe Roberts, Elementary Number Theory : A Problem Oriented Approach. First reason: the first third of the book is just problems, then the rest of the book is solutions. Second reason: the whole book is done in calligraphy. 
A: W.Sierpinski
Elementary Number Theory
From the master.
A: Another interesting book: A Pathway Into Number Theory - Burn

[B.B] The book is composed entirely of exercises leading the reader through
  all the elementary theorems of number theory. Can be tedious (you get
  to verify, say, Fermat's little theorem for maybe $5$ different sets of
  numbers) but a good way to really work through the beginnings of the
  subject on one's own.

A: I would still stick with Hardy and Wright, even if it is quite old.
A: I was shocked to see no one mentioned LeVeque's Fundamentals of Number Theory (Dover). He also authored Elementary Theory of Numbers with same publisher.
A: I like Niven and Zuckerman, Introduction to the Theory of Numbers.
A: For people interested in Computational aspects of Number Theory, 
 A Computational Introduction to Number Theory and Algebra - Victor Shoup , is a good book. It is available online.
A: Serre's "A course in Arithmetic" is pretty phenomenal.
A: I recommend Primes of the Form x2 + ny2, by David Cox.  The question of which primes can be written as the sum of two squares was settled by Euler.  The more general question turns out to be much harder, and leads you to more advanced techniques in number theory like class field theory and elliptic curves with complex multiplication.
A: Stewart&Tall's "Algebraic Number Theory" is great.
A: William Stein has shared his Elementary Number Theory online: http://wstein.org/ent/ It is accessible, lots of examples and has some nice computation integration using SAGE. I'll be using it this semester with secondary teachers, and will report back if things go particularly well or poorly with it.
A: Apostol, Introduction to Analytic Number Theory. I think it' very well written, I got a lot out of it from self-study.
A: There are many books on this list that I'm a fan of, but I'd have to go with Neukirch's Algebraic Number Theory. Great style, great selection of topics.
A: A concise introduction to the theory of numbers by Alan Baker (1970 Fields medalist) covers a lot of ground in less than 100 pages, and does so in a fluid way that never feels rushed. I love this little book.
A: Perhaps "best ever" is putting it a bit strong, but for me one of the best besides L E Dickson's books was "Elementary Number Theory" by B A Venkov, which does have an English translation. 
One advantage of this book is that it covers an unusual and quite eclectic mix of topics, such as a chapter devoted to Liouville's methods on partitions, and some of these are hard to find in other texts.
The best benefit for me, paradoxically, was that the English translation I worked with was littered with misprints, in places a dozen or more per page. So after a while it became quite an enjoyable challenge to find them, and this meant having to study and consider the text more closely than one might have done otherwise!
A: 
Books on Number Theory for anyone who loves Mathematics?  (Beginner to Advanced & just for someone who has a basic grasp of math)

New answer to an old question:
An Illustrated Theory of Numbers, by Martin H. Weissman ($2017$)
A: One of my colleagues, a number theorist, recommended the little book by van den eynden for beginners.  my favorite is by trygve nagell.  (I am a geometer.)  One of my friends, preparing for a PhD in arithmetic geometry?, started with the one recommended by Barry, Basic number theory.  As I recall it's for people who can handle Haar measure popping up on the first page of a "basic" book on number theory.
I also recommend Gauss's Disquisitiones Arithmeticae.
A: A Friendly Introduction to Number Theory by Joseph H. Silverman.  Although the proofs provided are fairly rigorous, the prose is very conversational, which makes for an easy read.  Also, the material is presented so that even a student with a low to moderate level of mathematical maturity can follow the text conceptually and do many of the exercises, but there are plenty of exercises to stretch the more curious mathematician's mind.  
As an undergrad I found it very useful and even years later it is one of my all-time favorite number theory references.
A: Elementary Number Theory - by David M. Burton if you want it somewhere halfway between fast and slow.
link
A: A Classical Introduction to Modern Number Theory by Ireland and Rosen  hands down!
A: It depends on the level.
For an undergraduate interested in algebraic number theory, I would strongly suggest (parts of) Serre's Cours d'arithmetique and also Samuel's Théorie algébriques des nombres.
For a graduate student aiming at a future of research work in number theory, Cassels & Fröhlich is a must.
A: Problems in Algebraic Number Theory is written in a style I'd like to see in more textbooks
A: Basic Number Theory by Andre Weil.  It's hard going and mind-blowing.
A: One of my favorites is H. Davenport's ${\bf The\ Higher\ Arithmetic}$
A: Kato's "Fermat's Dream" is a jewel. (Full disclosure: actually I saw it mentioned either here or on mathoverflow, and I was looking for the post to thank the source.)
A: My favorite is Elementary Number Theory by Rosen, which combines computer programming with number theory, and is accessible at a high school level.
A: For a highly motivated account of analytic number theory, I'd recommend Harold Davenport's Multiplicative Number Theory.
A: In my opinion, "the theory of numbers" by Neal H. Mccoy contains all number theory knowledge that a common person should have. 
A: Elementary Number Theory by Underwood Dudley. This is cheap being a dover book (so no big financial investment) and has witty & imaginatively written proofs, although it might be challenging to the complete beginner. Nevertheless, there are hints for a lot of the exercises.
A: For someone who wants to see a lot of neat Number Theory facts and isn't too interested in proofs, Albert Beiler, Recreations in the Theory of Numbers, is perfect.
