We Hindus perform many vratas-daily, weekly, monthly, annually. Most of these described and detailed in Puranas, Ithihasas and other scriptures.
Opening this question to know which vratas are there and where they are described to know more about the original narration, time when to perform, who should do, procedure and benefits.
Note: Aware that one answer here will not cover all the vraras but hoping the learned members here can contribute and we'll have this question as a library of Vratas
############################## Rishi Panchami (Ṛṣi-Pañcamī) ##############################
Source: Skanda Purana
Sanatkumāra narrates to Vyāsa about Rishi Panchami as below:
The wives of seven sages (sapta rishis), the chaste wives of the sages, got polluted due to the faults of Svāhā. They were defiled by Pāvaka (Agni). Hence they were forsaken by the sages. They wandered from forest to forest.
When a long time elapsed, Nārada of divine vision came into that forest desirous of doing something beneficial and pleasing to them. He was received and honoured by them. The sage who always observed sacred vows, sat there. Words befitting the occasion and the place were uttered by him blandly and sweetly. “What unnatural incident took place resulting in the humiliation of respectable ladies like you? You are chaste mothers of the world. Why were you abandoned by the sages?”
The wives of the sages replied: "O dear one, we do not know our fault on account of which we have been forsaken suddenly by the sages who maintain sacrificial fires. Arising in the context of Kārttikeya as bad luck would have it, something sinful originating from popular rumour has immediately come out. What shall we do? Where shall we go? What shall be the expiatory penance? What is the deity the propitiation of whom may bring in the merit enabling us to regain the proximity of our husbands? O holy Sir, tell us this decisively. You know everything accurately."
On being asked by those wives of the sages, Nārada meditated for a long time and said what should be conducive to their welfare.
Nārada asked them to go to Mahākālavana (Avanti) and take bath in Gayā Tīrtha. On hearing the words of Nāraḍa, the wives of the sages were prompted. Desirous of their own welfare, they came to that forest area in Mahākālavana where there was the Tīrtha called Gayā.
On being purified by going there, they performed holy acts like sacred bath, charity etc. On the fifth lunar day (Panchami) in the bright half (Shukla Paksha) of Bhādrapada called Ṛṣi (i.e. Ṛṣi-Pañcamī), a Vrata (called Ṛṣi-Pañcamī) was properly observed by those meritorious ladies.
They fasted for a night (i.e. day) and along with it kept awake at night too. As soon as the Vrata was performed by them they were rid of all sins instantly.
They shone well. They who had been degraded due to the anger of their husbands, immediately regained their Gṛahāśrama (the stage of a house-wife) offered by the sages along with the sacred fire as before.
Ever since then this Pañcamī was named after the sages as Ṛṣī Pañcamī. Men or women should perform Vrata on that day by remaining pure and mentally concentrated. They should take in only Nīvāra (wild rice that grows without being sown) as food. They will never have any adversity or unhappiness. Women will never become unfortunate (e.g. widow) or distressed. They shall never be separated from mothers, sons, wealth etc.