Why do people fast on Monday, Tuesday or Saturday?

each weekday is linked with a deity — Monday with Śiva, Tuesday with Hanumān, Saturday with Śani and Hanumān.

The seven weekdays are named after the seven visible grahas, and over time each became associated with a deity. Monday (Soma, the Moon) is Śiva’s day, since Śiva wears the moon on his head; the Solah Somvār vrata (sixteen Mondays) and Śrāvaṇa Mondays are very popular. Tuesday (Maṅgala, Mars) is widely kept for Hanumān, with sweet offerings and the Hanumān Cālīsā. Saturday (Śani) is kept to honour Śani-deva, often with sesame oil and black sesame, and also for Hanumān, who tradition says protects devotees from Śani’s hardships.

The usual form is one meal a day (often without salt in some vratas), a visit to the temple or home pūjā, and the relevant kathā.

These are devotional choices, not obligations. Pick one that draws you, keep it simply and regularly, and don’t take it up out of fear.

Monday–Śiva · Tuesday–Hanumān · Saturday–Śani · Source: common devotional practice

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