Someone or something that is solemn is very serious rather than cheerful or humorous.
His solemn little face broke into smiles.
他那严肃的小脸突然绽放出笑容。
A solemn promise or agreement is one that you make in a very formal, sincere way.
She made a solemn promise to him when they became engaged that she would give up cigarettes for good.
他们订婚时,她向他许下郑重的承诺,说她会永远戒烟。