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In Hindu traditional calender Their are 12 months (मास)
Chaitra (चैत्र), Vaisakha (वैशाख), Jyaistha (जेष्ठ), Asadha (आषाढ), Sravana (श्रावण), Bhadrapada (भाद्रपद), Asvina (अश्विन), Kartika (कार्तिक), Agrahayana (अग्रहायन), Pausa (पौष), Magha (माघ), and Phalguna (फाल्गुन).
Who have first given the names to Months?
What is the meanig of each Month's name?
Are their any stories related to these Names?
These 12 months are derived from the name of nakshtras. As I discuss in my answer here, the names of nakshtras are itself stated in Atharva Veda. In a particular month there is high probability of occurring full moon in particular nakshtra. Of course it may not occur exactly but it occurs near.
On the basis of this 15 nakshtras were choosen and months were named on the basis of that name. For eg. There is high probability of occurring full moon in or near Visakha nakshtra in Vaisakha and so that full moon is termed as Vaisakh Poornima and the month is named Vaisakh.
So, the months are derived from these nakshtras:
Chitra Nakshtra = Chaitra month
Visakha Nakshtra = Vaisakha month
Jyestha Nakshtra = Jyestha month
Purva Aashaadha, Uttara Aashaadha Nakshtras = Aasadha month
Sravana Nakshtra = Sraavana month
Uttara Bhaadrapada, Poorva Bhaadrapada Nakshtras = Bhadra month
Asvini Nakshtra = Asvina month
Krittika nakshtra = Kaartika month
Mrigashira nakshtra = Maarghasira month
Pushya nakshtra = Pausa month
Maghaa nakshtra = Magha month
Uttara Phalguni, Purva Phalguni nakshtras = Phalguna month
The full moon either occurs in those nakahtras or near those nakshtras of a particular month and hence these months derive their name from those nakshtras.
Months chaithra,vaishakha,...,phalguna in lunar calender is named because of the position of moon after one lunar cycle ~29.5days(purnima) as mentioned in the previous answer.
In hindu calender, we used to follow solar calender as well. Here, months may be named as mesha,rishabha, ...,meena. It means when the surya graha is it starting point of the rashi ie. Every 30degree(~30days) - 0,30,60,60...,360 as mesha sankranthi, rishabha sankranthi, mithuna sankranthi and so on.
