I agree with the leading poster. One can't really answer this in a few paragraphs. To really get a complete answer you have to study modular forms in detail and you will see why they are so enchanting. I will give my partial answers to questions 1,2.
(Answer to Question 1) They are interesting for two reasons. First, a modular form satisfies so many functional identities their existence almost seems unreal. There is so much structure involved with Modular forms that one can prove beautiful results. Second, modular forms are deeply connected with several number theoretic objects.
(Answer to Question 2) The easiest application of modular forms to understand is in classic analytic number theory. Modular forms often act as generating functions for several interesting arithmetic functions. I will give three examples: 1) the eta function almost generates for the number of integer partitions of $n$, 2) Powers of theta functions generate the number of ways a number can be written as a sum of squares, 3)Eisenstein series generate weighted divisor functions.
Knowing these generating functions are modular allows for extremely precise approximation of such generating function, and by Fourier analysis, we can obtain information about the behavior of the arithmetical function you are studying.
