The Alphas and the Betas represent the highest class of people who have been created for this new world. Huxley uses the Greek alphabet to name them and Alpha and Beta are the first two letters of the Greek Alphabet therefore designating their place in the society as first and second. They are the most intelligent people in the society - the Alphas being smarter than the Betas. The Alphas and the Betas associate with one another whereas, you would not find an Alpha having much social interaction with an Epsilon for example.
The Alphas and the Betas rule society. The main characters are usually Alphas with the exception of the women who work in the fertilization factory or Hatchery. They are Betas. This shows us that the society is primarily run by men.
The author mentions them in the novel so frequently because they are our guides through this society. We see the New World through their eyes and since they are the most intelligent people in the world - they can also explain their world to the reader. Since a lower individual like a Delta or Epsilon would not have the understanding of the world like an Alpha - to see the world through their eyes would be limiting.