Short answers, pulled from the story.
Amitābha is a compound of the Sanskrit words amita, meaning "without bound" or "without limit," and ābhā, meaning "light" or "splendor." The name therefore means "boundless light" or "limitless light." Amitābha also goes by the name Amitāyus, which combines amita with āyus, meaning "life," and translates as "limitless life."
Sukhāvatī, meaning "Utmost Bliss" or "Blissful," is the pure land created by Amitābha Buddha, located billions of worlds away in the western direction. According to the Sutra of Measureless Life, rebirth there is granted to beings who sincerely entrust themselves to Amitābha and think of him even ten times with genuine faith. The Smaller Sukhāvatīvyūha Sūtra specifies that holding Amitābha's name in mind with undistracted focus for one to seven days is sufficient.
The forty-eight vows are promises made by Amitābha, then still a bodhisattva monk named Dharmākara, pledging that unless each condition was fulfilled he would not attain Buddhahood. They set out the nature of the pure land he intended to create, the conditions for rebirth there, and the qualities beings would enjoy. The eighteenth vow, stating that any being who sincerely thinks of Amitābha even ten times shall be born in his land, is called the "original" or "fundamental" vow in East Asian Pure Land Buddhism.
The earliest known physical evidence is a statue found in Govindnagar, Pakistan, now at the Government Museum in Mathura, dated to 104 CE. It was dedicated to "Amitābha Buddha" by a family of merchants during the Kushan Empire. Kenneth Tanaka places Amitābha's emergence as a central figure in Gandhāran Buddhism in the first century CE.
Tibetan Buddhism treats them as two distinct figures, unlike East Asian Buddhism where both names are used interchangeably. Amitābha is depicted in simple monk's clothing and is understood as a manifestation body Buddha. Amitāyus is depicted in ornate robes, a crown, and jewels as an enjoyment body, and is specifically associated with longevity practices. Amitāyus is one of the three deities of long life in Tibetan Buddhism, alongside White Tara and Usnisavijaya.
Hōnen was a Tendai monk who lived from 1133 to 1212 and was the most influential figure in the Japanese Pure Land movement during the Kamakura period. Converted to the Pure Land path through the writings of the Chinese master Shandao, Hōnen argued that practitioners should set aside all other Buddhist practices and rely solely on the recitation of Amitābha's name. The traditions his followers established, including Jōdo-shū and Shinran's Jōdo Shinshū, became the largest Buddhist tradition in Japan and remain so today.