The monthly Chaturti day that comes after full moon (or what is known as "thei-pirai" or "Krishna Paksha" Chaturti) is called Sankashti (aka Sankatahara) Chaturthi. The Chaturthi that comes after new moon (or "valalar pirai" or "sukla paksha") is known as Vinayaki Chaturthi. Ganesha Bhaktas observe a fast on these days: it begins at Brahmamuhurta and ends after moonrise.
The scriptures say that those who fast on these days gain material progress, happiness and fulfillment of desires. It also helps to overcome problems and tribulations in life. People who are suffering from troubles caused by the bad effects of Mars and Moon would find relief from these. In one of the Puranas Lord Krishna explains to Yudhishtira, the eldest of the Pandavas the importance of this vrata. It is also similarly extolled in Narasimha Purana and Bhavishaya Purana.
Swami Sivananda explains the significance of Kartigai vrata in His “Shanmukha and His Worship” (pg. 8): “……there are some Vratas, or observances. There was once a king, Muchukunda by name. He was a staunch Saivite. He approached Sage Vasishtha to advise him on the importance and methods of observing some Vrata by which he could gain personal success, peace and prosperity. Vasishtha indicated the Vratas of Lord Skanda and detailed their significances.
“One of them is the Vrata observed on every Friday. By observing this Vrata of Friday for three years, Bhagiratha overcame his enemy Gora. On Friday, after a clean bath, one should do Puja and Archana for the Lord with extreme devotion. He who does this is bound to achieve success in all his undertakings.
“Another Vrata is the Karttikai Vrata. On every Karttikai (3rd constellation of the 27 stars) day, this Vrata is to be observed with faith and devotion, and with external purification by a bath, before the Puja, etc. This Vrata was observed by Narada Rishi as per the advice of Lord Ganesa. After a period of 12 years, Narada gained his ambition, viz., the supreme position among the great Sapta Rishis.”
Guruvaaram (aka Guruvara)
Guruvar — var meaning "day" and Guru refers either to Brihaspathi, the guru to the gods (devas) or to the planet Jupiter. Generally, the day is regarded very auspicious for the worship of one's Diksha Guru.
According to Hindu traditions, Thursdays are regarded as auspicious to start any educational endeavours. It is on Thursdays that special worship services are offered to Dakshinamurthy in many Saivite temples. Some temple traditions hold full moon nights, particularly the night of the Guru Poornima as the appropriate time for worship services to Dakshinamurthy.
Swami Sivananda says, “Thursday or Guruvara is the sacred day for a disciple to worship his Guru.” (pg. 67, Guru Bhakti Yoga).
Swami Sivananda and Paramahansa Yogananda have said that one should observe fast once a week. This is over and above the monthly fast, e.g., Ekadasi, etc. Thursday is a good day to fast. .
Incidentally, the followers of Judaism and Islam consider Thursday an auspicious day to fast.