What structure in the female reproductive system produces eggs?

ovary

These oocytes are formed during the early development of the person, it is together formed in the embryonic stage of development.

In puberty, these oocytes have now been mature thus, the individual is ready for her first menstruation, it takes time before these oocytes are fully developed but they are formed as early as the state of the genitalia, brain, body was formed.
Somehow, it is like the brain, the brain is slowly developing til the near age of puberty or late childhood. 
Eyes are also an analogy to oocytes, during infancy these organs aren't developed yet and these years the child can only see vague and blurred vision and takes almost 2 years for the child to see clearly.