Hindus observe Nava Ratris twice in the same calendar year. One is associated with Rama Navami (ie birth of Lord Rama) and the other is with Lord Rama's victory over Ravana. But why do we worship the nine forms of the Devi on both the occasions? What is the logic and theory behind the same?
First of all, Navaratri refers to the worship of the nine forms of the Mother Goddess, one form each day in the following order:
These forms are described in the Devi-kavacha of the Devi-Mahatmya (slokas 3-5).
Second, in the Devi-MahAtmya, Devi Herself mentions that She is worshipped twice in a year :
saratkAle mahApujA kriyate yA cha varshiki (chapter 12, mantra 12), meanings 'My great Annual great worship (Chaitra) and also done in the Autumn (Sarat-kAla)'.
It is believed that all the devatAs remain asleep during the DakshinAyana. It was because of the killing of RAvana, BrahmA awakened Devi during the Autumn. This is therefore known as the AkAla-vodhana (Untimely Awakening of Devi). Here, the performance of the awakening ceremony (vodhana) is required.The Mantra of the Bodhana reveals this:
Om Aim rAvanasay vadhArthAya rAmasyAnugrahAya cha/akAle brahmanA bodho devyas tvayi kritah purA//
The story of Sri Rama's worshipping of Devi as narrated by BrahmA can be read here: http://www.kamakoti.org/kamakoti/details/devibhagvatpurana28.html
The Chaitra-MahApujA is done during the UttarAyana and so no vodhana is done then.
In most cases however, the worship starts on the Sasthi tithi of the sukla paksha and this is known as the 'sasthayAdi-kalpArambha', especially in Bengal.
As Rickross points out in his answer (Significance of Chaitra Navratri?) quoting from the Devi-Bhagavatam, both of the periods are dangerous and so Devi's worship is advised in the Devi-BhAgavatam to get Her blessing:
3-5. Vyâsa said :– O king! Hear about the vow of auspicious Navarâtra. This has to be performed with loving devotion in the vernal season; but its special season is autumn. The two seasons, autumn and spring, are famous as the teeth of Yama, the God of Death; and these are the two seasons, very hard for the persons to cross over. Therefore every goodfaring man should everywhere perform this vow very carefully.
6-8. O king! The people are very much afflicted with various terrible diseases in these two seasons autumn and spring and many lose their lives during these portions of the year. Therefore the wise should unquestionably worship with great devotion the Chandikâ Devî in these auspicious months of Chaitra and Âs’vin.
There is also a Gupta NavarAtri worshipped in the sukla-paksha of Magha (Reference: https://www.prokerala.com/festivals/magha-gupta-navratri.html) but that is meant mainly for the spiritual aspirants and not celebrated as a festival.