The story
Ikigai (生き甲斐) is Japanese for what makes a life worth living. The first part comes from ikiru, to live; the second is kai, worth or value, which softens to gai inside the compound. In Japan it is an ordinary word, used for ordinary things. A grandchild can be someone's ikigai, and so can a garden or a job.
The word was given wider attention in Japan by the psychiatrist Mieko Kamiya, who worked with leprosy patients at the Nagashima Aiseien sanatorium and in 1966 published a book on the subject, Ikigai ni tsuite, On the Meaning of Life. Later researchers distinguished between the source of ikigai, the person or activity, and the feeling that your life has value because of it.
Outside Japan the word took a different turn. The diagram that travels with it online, four overlapping circles for what you love, what you are good at, what the world needs and what you can be paid for, did not come from Japan. It was drawn in the West, probably after an earlier diagram about purpose, and critics point out that it sets a bar most lives never reach. The Japanese idea is less demanding: nobody needs a career to have an ikigai.
English usually says purpose, or reason for living, or borrows the French raison d'être. Those come close, but they sound like something you must find. In Japanese, ikigai can be as small as the plants on a balcony that need watering every morning.