The three things in the title are examples of how we form compound words.

new moon is the open form
   post office, real estate, half sister, attorney general, grade point average

firefly is the closed form
   secondhand, softball, keyboard, notebook, butterfly, workmanship

six-year-old is the hyphenated form
   daughter-in-law, over-the-counter, mass-produced, one-eighth, twenty-three

Which form should you use? Check an authoritative dictionary.

To further confuse the issue, the following are all corrrect:
   My six-year-old son.
   My son is six years old.
   He is a six-year-old.