Why are prime numbers important in real life? What practical use are prime numbers? Why do we emphasise the teaching of prime numbers?
 A: For real life applications see Real-world applications of prime numbers?. Why are prime numbers so important that we teach about prime numbers ? This question is not so easy as it seems.
Of course, for elementary number theory, prime numbers are like the "atoms", and several questions involve prime numbers. 
For mathematics in general, the value of prime numbers lies much deeper. For example, the distribution of prime numbers encodes very deep mathematical information in general (not only via the Riemann Hypothesis). Completions of the rational numbers naturally lead to $p$-adic fields, and
the idea of being "prime" applies to many other structures (like prime ideals, prime geodesics etc.).
So we emphasise teaching prime numbers because they lie at the very heart of mathematics.
A: Primes are of prime importance in cryptography (take for example the RSA cryptosystem). See http://en.wikipedia.org/wiki/RSA_(algorithm)
However one does Mathematics simply for doing Mathematics. What Mathematicians think today, may find its application even more than 200 years later. Although many branches (like variational calculus) came up from practical issues, not every branch of mathematics can be even remotely related to some real-life problem. 
Prime Number Theory is a very interesting topic. It not only consists of some exciting results and findings over the ages but it actually enhances our mathematical thinking and imagination.   
