I am fascinated by the area of Integer Partitions. A partition is a natural object: it answers the question "In how many ways can I split up a positive integer n into a sum of smaller positive numbers?" Partitions have connections to a wide variety of fields like Combinatorics, Number Theory, Hypergeometric Series, Lie Algebras, Statistical Physics, and many more. As with many areas in mathematics, one of the earliest drivers of research in partitions was Leonhard Euler, and more recently, it received a fillip through Ramanujan's fundamental contributions in the first two decades of the 20th century and whose impact continues to be felt to this day. Two books for a good introduction are:  

• Number Theory in the Spirit of Ramanujan by Bruce Berndt
• Integer Partitions by George Andrews and Kimmo Eriksson