The noun 案子 (ànzi) means "case," typically referring to a legal or criminal case. It is a colloquial equivalent of the more formal 案件. For example, 这样的案子 means "a case like this."
警察正在调查一个很复杂的案子。
Jǐngchá zhèngzài diàochá yī gè hěn fùzá de ànzi.
The police are investigating a very complicated case.
这个案子已经拖了三年了。
Zhège ànzi yǐjīng tuō le sān nián le.
This case has been dragging on for three years.
他一辈子办过很多大案子。
Tā yī bèizi bàn guò hěn duō dà ànzi.
He handled many major cases in his lifetime.