生き甲斐 is whatever makes a person's life feel worth living — it can be a person, an activity, or a role. Grammatically it behaves as a plain noun: Xが生き甲斐だ identifies what that thing is for someone. It is formed from 生きる plus the 〜甲斐 suffix seen in やり甲斐.
Common expressions
- 生き甲斐を見つける (to find one's purpose in life)
- 生き甲斐を感じる (to feel a sense of purpose)
- 生き甲斐を失う (to lose one's reason for living)
- 生き甲斐のある人生 (a life with purpose)
Cultural note
The four-circle diagram popular in English — the overlap of 好きなこと (what you love), 得意なこと (what you are good at), 必要とされること (what the world needs), and 稼げること (what you can be paid for) — is a foreign elaboration. In ordinary Japanese use the word is far humbler and carries no requirement that the thing be productive or paid: a grandchild or a garden qualifies.
Writing
Often written いきがい in hiragana. The word has been borrowed into English as 'ikigai,' usually in the four-circle sense rather than the everyday Japanese one.