Short answers, pulled from the story.
Buddhānusmṛti means Buddha-mindfulness, also translated as remembrance, commemoration, recollection, or mental contemplation of the Buddha. It is one of the various recollections, or anusmṛti, taught by the Buddha in the sutras.
The Buddhānussati Gatha lists nine epithets of the Buddha known as the Nava Guna: perfected, a fully awakened Buddha, accomplished in knowledge and conduct, holy, knower of the world, supreme guide for those who wish to train, teacher of gods and humans, awakened, and blessed. The verse is widely chanted in Theravada Buddhism and used in the Visuddhimagga.
Vakkali renounced the world out of devotion to the Buddha's physical appearance and despaired when the Buddha kept his distance during a rains retreat. The Buddha then projected an image of himself to Vakkali atop Vulture Peak, and Vakkali achieved Arahantship as he leapt into the air in joy.
In East Asian Buddhism, Buddhānusmṛti is called Nianfo, or Nembutsu in Japanese, and centers on reciting the phrase Namu Amida Butsu, meaning Homage to Amitābha Buddha. Following the influence of the 7th century patriarch Shandao, vocal recitation became more common than visualization in Pure Land traditions, including those of Hōnen and Shinran.
In Vajrayana Buddhism, deity yoga, known as Devata-yoga in Sanskrit, is treated as a tantric form of Buddhānusmṛti that adds a mandala image and mantra recitation to visualization of a chosen deity, or Yidam. According to Shangpa Rinpoche, deity yoga is the most common type of meditation practiced in Vajrayana Buddhism.
Early Buddhist sources focus mainly on Gautama Buddha, but later Mahayana traditions such as Pure Land Buddhism and Vajrayana also teach meditations focused on Buddhas including Amitābha, Maitreya, and Vairocana.