Padmini Ekadashi is a rare one. It belongs to the extra month, the Adhika masa, that the lunar calendar adds about once every three years to keep the months in step with the seasons, and so it comes only in the years that have one. The extra month is also called Purushottama masa, the month of Vishnu, which is why its Ekadashis are held in high regard. Padmini is also known as Kamala Ekadashi, both words meaning the lotus.
Padmini Ekadashi belongs to Adhika Jyeshtha, the extra month the lunar calendar adds about every third year to keep it in step with the seasons (see Adhika Masa). It is in the Shukla, the waxing paksha. This site labels it Adhika Jyeshtha in both the amanta and the purnimanta reckoning, but regional purnimanta calendars do not all name the dark half of an extra month the same way, so check your own.
The katha
The katha of Padmini Ekadashi is attributed by the tradition to the Padma Purana and the Bhavishyottara Purana; regional and printed tellings agree on the shape of the story and differ in its details. The story of this Ekadashi is about a king and queen who wished for a child.
King Kritavirya of the Haihaya line had many queens, but none had given him a son. He and his ministers went to the Gandhamadana mountain and performed austerities for a very long time, but nothing came of it. His chief queen, Padmini, born in the family of Ikshvaku, went to the sage Anasuya, the wife of Atri, and asked what she could do.
Anasuya told her of the extra month, called Purushottama, the month dear to Vishnu, and of its Ekadashi in the bright half. Padmini kept the fast with great sincerity, staying awake through the night with prayer and singing, and offered the merit to Vishnu.
The tradition says Vishnu appeared and granted her a son who would be a great ruler, and that this son was Kartavirya Arjuna, the thousand-armed king whose story is told in other books. The tale is used to say that the extra month is not empty time, but one that is especially suited to devotion.
How the fast is kept
In the extra month many families take on small extra vows, such as reading a sacred text, giving charity or keeping up daily worship. Padmini Ekadashi is the high point of the bright fortnight. The usual Ekadashi worship is done, and many people stay up through the night, singing and listening to the katha.
Because the Adhika month is not used for marriages and other first-time ceremonies in most traditions, it is regarded as a month for worship, charity and pilgrimage. Please read the Adhika Masa page for the rule behind it.
Which day counts as Padmini Ekadashi depends on the school. Smarta practice, which the table follows, keeps the day when the tithi is running at sunrise; Vaishnava practice often rejects an Ekadashi that is mixed with Dashami before dawn and moves Padmini Ekadashi on by a day. The parana comes after sunrise on the next day, within Dwadashi and clear of Hari Vasara, and the Timing column shows the window for each city. On Padmini Ekadashi, how much you eat matters less than how you spend the day. If you need food or water for your health, take it without guilt, and check with your doctor before changing medicine or diet for any vrat.
Where it sits in the year
Padmini Ekadashi is followed two weeks later by Parama Ekadashi, and the regular round resumes with Nirjala Ekadashi. It comes after Apara Ekadashi. The Ekadashi page lists all the named Ekadashis. For Padmini Ekadashi the practical steps are in the Ekadashi vrat vidhi, and the meaning of the tithi is on the Ekadashi tithi page.