Short answers, pulled from the story.
Other power (Chinese: tali, Japanese: tariki) refers to the power of a Buddha to inspire, sustain, and carry living beings to the Pure Land, where becoming a Buddha is made easy. It is contrasted with self power, which is the attempt to achieve enlightenment through one's own efforts. The scholar Mark L. Blum describes it as "something 'received' or 'influenced' from the sacred world beyond the self."
Tanluan (476-542) was the first Chinese thinker to apply the term other power to a Pure Land context. He described other power as the dominant causal condition for attaining Buddhahood, comparing it to a lame man who travels a thousand li in one day by the power of a boat, not his own effort.
Shinran (1173-1263), the founder of Jodo Shinshu, defined other power as being free of any form of calculation, a total entrusting to the primal vow of Amitabha Buddha without one's own working. He called the effortless and natural unfolding of the Buddha's power jinen-honi, which occurs when all self-judgment and conceit have been dropped.
Sympathetic resonance (ganying) describes the interaction between a practitioner's effort and the Buddha's responding power. Yunqi Zhuhong (1535-1615) compared it to a plucked lute string causing a nearby string to resonate. When a practitioner recites Amitabha's name with sincere intent, the Buddha responds and the two minds attune to each other.
Indian Mahayana Buddhism contained several concepts that preceded East Asian other power, including buddhanubhava (the Buddha's majesty and causal power to influence others), buddhaadhisthana (the Buddha's sustaining and supporting force), parinaamana (the transference of merit), and pranidhaana (the power generated by a bodhisattva's past vows).
Honen (1133-1212) described other power as having implicit faith in the repetition of the nembutsu without looking back on one's virtuous or vicious deeds. He illustrated it with the image of a large boulder placed on a ship reaching the far shore not by its own mobility but by the ship's power. Because the path relies on the Buddha's power rather than personal effort, Honen argued it is available to all kinds of people, not only monks or elite practitioners.