Agni Nakshatra is the period of the Sun travelling in the star of Bharani (3 &4 padams ), Karthigai (all 4 padams) and Rohini (1 padam).
This period is called Agni Nakshatra because Agni is the athidevata of the star Karthigai which is a constellation of six stars who are born of the third eye of Lord Shiva. Since Agni is the Athidevata of this constellation, the Nakshatra is called Agni Nakshatra. When the sun passes through this constellation, it is naturally very hot.
The time is considered auspicious for worshipping Lord Muruga since Karthigai is the birth star of Muruga and he is the karaka for gnaana. By burning the fire of karma through his worship, we are able to light our inner fire namely gnaana. During Agni Nakshatra period, people collect water from holy rivers like Cauvery for anointing Muruga. They collect this Teertha as prasadham for them. Teertha kavadai during this time reduces the impact of heat on devotees.
The time is not considered auspicious because the Yama, the Lord of Death is the athidevata of the star Bharani and thus the time is not propitious for conducting auspicious events. The effect of the hot sun may not bring positive bearings upon the people. So, people avoid auspicious events during the period of Agni Nakshatra.