In China, deciding to 报案 (bào'àn) is often a more significant decision than it might be in some Western cultures. There can be a cultural reluctance to involve authorities to avoid `麻烦 (máfan)`—trouble, hassle, or inconvenience—for oneself and others. The act of formally reporting a case signals that an incident is serious enough to warrant official intervention and can't be resolved privately or ignored. A crucial cultural and linguistic distinction is between 报案 (bào'àn) and 报警 (bàojǐng).
In essence, you might `报警` on the phone first, and then the police might ask you to come to the station to `报案` and provide a detailed statement.
报案 (bào'àn) is a formal term used in specific situations. It's not for trivial complaints.
The most common mistake for learners is confusing 报案 (bào'àn) with 报警 (bàojǐng).
In short: See a crime in progress? 报警 (bàojǐng). Realize your bike was stolen last night? Go to the station to 报案 (bào'àn).