季军 (jìjūn) is used in formal contexts related to the results of a competition. It is standard vocabulary in media, official announcements, and sports commentary.
Sports: This is the most common use case. You will constantly hear it during the Olympics, the World Cup, national games, and even local school sports meets.
Other Competitions: Its use extends to any formal contest with a clear ranking system, such as singing competitions, academic Olympiads, chess tournaments, and e-sports championships.
Formality: While 季军 is the formal title, in very casual conversation, a speaker might simply say someone was 第三名 (dì sān míng), meaning “number three.” However, on a certificate, on the news, or when conferring the actual honor, 季军 is the correct and respectful term to use.