Why the eleventh day specifically
Ekadashi falls on the eleventh day of both the waxing and waning lunar fortnights, giving 24 Ekadashis a year (occasionally 26 in a leap lunar year). Puranic tradition holds that observing this fast pleases Vishnu directly and helps purify both body and mind — the number eleven is linked in some interpretations to the ten senses (five of action, five of perception) plus the mind, all of which the fast is meant to bring under discipline.
How the fast is actually kept
Observance varies by how strict a devotee wants to be. The most common form avoids grains and beans entirely, permitting fruits, milk, and certain vegetables. Stricter observers keep a nirjala (waterless) fast, eating and drinking nothing at all until the following morning. Because a full waterless fast is demanding, it's most associated with a single Ekadashi each year — Nirjala Ekadashi, in the summer month of Jyeshtha — which is considered powerful enough to substitute for observing all 24 Ekadashis in that year if kept properly.
Named Ekadashis through the year
Each Ekadashi carries its own name and story: Devshayani Ekadashi marks the beginning of Chaturmas, when Vishnu is said to go into cosmic sleep; Prabodhini Ekadashi marks his waking, four months later; Mokshada Ekadashi, in the month of Margashirsha, is traditionally associated with the day the Bhagavad Gita was spoken on the battlefield of Kurukshetra. Regional calendars sometimes name and date these slightly differently.
Breaking the fast (Parana)
The fast is broken the next morning, Dwadashi, within a specific window of time — breaking it too early or too late is traditionally considered to reduce the fast's merit. Many devotees attend a Vishnu or Krishna temple on Ekadashi itself, and use the day for extra time in prayer, reading scripture, or japa rather than routine work.