Sankashti Chaturthi

Sankashti Chaturthi, also known as Sankatahara Chaturthi and Sankashti, is a holy day in every lunar month of the Hindu calendar dedicated to the Hindu god Ganesha. This day falls on the fourth day of the Krishna Paksha. If this Chaturthi falls on a Tuesday, it is called Angaraki Sankashti Chaturthi, Angaraki Chaturthi, Angaraki and Angarika. Angaraki Sankashti Chaturthi is considered highly sacred.
This is said to have started around 700 BC as an obstacle removal ritual regarding conflicting views of confidence as stated by Abhisheka Maharishi in tutoring his pupil Aishwarya while deriving due reason from the scriptures.

Practices

Sankashti Chaturthi comes on every fourth day after Pournami full moon of the Hindu lunar calendar month.
On Sankashti Chaturthi, devotees observe a strict fast. They break the fast at night after having darshan of the moon preceded by prayers to Ganesha. The Angaraki Chaturthi. On the Sakashta Chaturthi day of each month, the 'Sankashta Ganapathi Pooja' prayer is performed. Each Vrata has a purpose and is explained to us by a story known as the Vrata Katha. This prayer offering has 13 Vrata Kathas, one for each month and the 13th story is for adhika. The story pertaining to that month is recited.
The Sankasta Ganapathi Pooja – 13 names and peetas

Month	Name of Ganesha of which the pooja is performed	Name of the peeta
Chaitra	Vikata Maha Ganapati	Vinayaka Peeta
Vaishakha	Chanakra Raja Ekadanta Ganapathi	Srichakra Peeta
Jeshtha	Krishna Pingala Maha Ganapati	Sri Shakthi Ganapathi Peeta
Ashadha	Gajaanana Ganapati	Vishnu Peeta
Shravana	Heramba Maha Ganapati	Ganapathi Peeta
Bhadrapada	Vignaraja Maha Ganapati	Vigneshwara Peeta
Ashwayuja	Vakrathunda Maha Ganapati	Bhuvaneshwari Peeta
Karthika	Ganadipa Maha Ganapati	Shiva Peeta
Margashira	Akuratha Maha Ganapati	Durga Peeta
Pushya	Lambodara Maha Ganapati	Soura Peeta
Magha	Dwijapriya Maha Ganapati	Samanya deva Peeta
Phalguna	Balachandra Maha Ganapati	Agama Peeta
Adhika	Vibhuvana Palaka Maha Ganapati	Doorva Bilva Patra Peeta

In North and Central India, this fast is observed on the fourth day of Krishna Paksha of Magha month of the Hindu calendar. Durga women perform Arghya to Chandrama at night. Bhugga and radish are donated and eaten to complete the fast. It is one of the main rituals celebrated by the Lodhi Rajputs.

Angarki Sankashti Chaturthi

Angarika Chaturth is a Sankashti Chaturthi falling on Tuesday. It is considered highly auspicious among all Sankashti Chaturthi days.
Angaraka, the son of the earth goddess Prithvi, was an accomplished rishi and a great devotee of Ganesha. He worshipped Ganesha and sought his blessings. On Magha Krishna Chaturthi, Ganesha blessed him and asked him for a wish. Angaraka expressed that his only wish was to be associated with Ganesha's name forever. The deity granted his wish and proclaimed that whoever worships Ganesha on Angarika Chaturthi will be granted all that he/she prays for. From that day onwards, Magh Krishna Chaturthi came to be known as Angarak Chaturthi.
On the day of Angarika Sankashti Chaturthi, the devotees observe a strict fast from morning till evening. They break the fast at night after having a darshan/auspicious sighting of the moon, preceded by prayers and a pooja for Ganesha. The Angarika Chaturthi devotees believe their wishes will be fulfilled if they pray on this auspicious day. The fast of Sankashti Chaturthi is generally started from the day "Angarika Sankashti Chaturthi". Also Angarika Sankashti means deliverance during troubled times, hence observing this fast is believed to reduce a person's problems, as Ganesha is the remover of all obstacles and the supreme lord of intelligence. Before moonlight, the Ganapati Atharvashesha is recited to summon the blessings of Ganesha.