Image source : “Provided by the City of Fukuoka.”
First of all, a policeman could ask you anytime to check your identity, so you are supposed to always have your passport or an identity document with you.
When calling the police?
If you are involved a traffic accident, call the police to the 110 (free dial).
Even if the case you think you do not need the presence of the police and could directly deal with the third party, you should call the police. In the case you don't, insurance may refuse to recognize the accident.
Once you reach the police, tell them:
What happened, where it happened, and when it happened. Also they will ask you your name and a phone number.
Same if you are witness or victime of a crime, call the police on 110.
If you are victim of theft, you can call the 110 to report the theft or go to the nearest police station.
If you have lost or left your personal items somewhere, you can report the loss to the police station. You might be surprised how often you could find it back!
If you found a lost item, bring it to the nearest police station. Of course if you found it in a train, subway, department store or taxi, you should inform the place staff, so they could bring it to the Lost and Found section.
Usually japanese people bring back lost items when they find one. A colleague explained me that even when his son found a ¥100, they went together to the police station, and register this coin as a lost item. In the case no one ask for the lost of this ¥100 coin, the person who bring it back (my colleague's son) could receive it in the following month (6 month I think).
Of course, no one is going to ask for a ¥100 lost, and many people may think it is a waste of time to go especially to the police station, but I personnally think it is awsome! Educate children to bring back anything they found to the police station, whatever the value, generate an automatism to bring back a lost item when they will find one.
Find back your lost articles is probably one of those Japan's specificities that you might enjoy if you have the opportunity to visit Japan.