Panchaagni is a compound word - Panch + Agni. Panch means five; Agni means fire. Panchaagni means Five types of Agni (fire or heat).

(1) People who tolerate Panchaagni during their Tap normally burn fire around them in four dirctions and sit under the Sun even in scorching heat of Summer. Thus they are surrounded by Panchaagni.

(2) For performing Vaidik sacrifices, consecrated fires are necessary. Five such fires, the maximum number, are
1. Gaarhapatya Fire - this Agni live in houses and is used to cook food; and in stomachs of living beings to digest food
2. Aavaahneeya Fire - This Ani appears as Soorya
3. Dakshinaagni, (Anvaahaaryapaachan) - Atmospheric Agni in the form of lightning
4. Sabhya, and
5. Aavaasatya Fire

(3) The great scholar and Minister Vidur is rendering his learned treatise on statecraft and lifestyle management to his King and elder brother Dhritaraashtra - "O bull of Kuru race, even man should serve with due diligence and sincerity the following five fires. They are (1) his father (2) his mother (3) the fire itself (4) his own pure inner self (5) the preceptor."

Panchaagni Vidyaa
There is another usage -- The Panchaagni Vidyaa - Brihadaaranyak Upanishad (6.2) and the Chhaandogya Upanishad (5.3.10) mention about this. The five Fires mentioned in these are-
1. Dyu Lok (Sky)
2. Parjanya (Rain)
3. Lok (This world)
4. Purush (Man)
5. Yosh (Woman)

This Panchaagni Vidyaa was taught by king Pravaahan to the sage Uddaalak Aaruni also known as Goutam as the five fires.