A somewhat older or formal term for a pilot or aviator; in modern Japanese, パイロット is more common.
彼は若い頃、飛行家になることを夢見ていた。
When he was young, he dreamed of becoming an aviator.
Compound of 飛行 (hikō, 'flight') and 家 (ka, 'person who does something professionally'). The exact historical origin is uncertain, but it follows a common pattern for occupational nouns.