A great introduction to (surprise!) modern Number Theory, by Kenneth Ireland and Michael Rosen, and published in Springer-Verlag's outstanding Graduate Texts in Mathematics. It is the textbook of choice for many basic number theory courses. It is commonly referred to as "Ireland and Rosen", as you are expected to know all about it (and the title is just too long and unspecial)...
The book assumes the reader to have some familiarity with abstract algebra, and is probably appropriate for an advanced undergraduate course, or a basic graduate one. It starts from absolute basics (e.g. unique factorization) and goes on to give a fair idea (anyway, to the extent I can follow) of most basic topics and some of what number theorists get to do nowadays. Given some prior knowledge, it makes for good reading, and there is a fair amount of exercises to make sure you're following and introduce a few further concepts. There is some attempt to put things in historical perspective, e.g. how more abstract ideas about rings and modules arise from an effort originally aimed to study the integers through prime numbers et cetera. | 
