Influence is also called 势 (shi4) or 势力 (shi4 li4) in Chinese[1], 势力 (seiryoku) or (gaisei) in Japanese and ? in Korean.
Each stone has a certain influence on the board. Captured stones have an influence close to zero. Surrounded stones that make a live group, influence only the surrounding group. Stones that are in open space have an influence on that open space and the stones bordering that open space. Influence is a long-range effect. For example: a ladder breaker changes the balance in the opposite corner.