The ordinary municipal Police are more like traffic wardens than officers of the law. They don't usually carry guns but as said above, the local Mayor can authorise them to carry. But are they trained to use guns? Probably not.
There are two National Police forces - the Police mainly in towns, the Gendarmarie mainly in rural areas. Both are semi-military and usually armed.
The CRS are the Police Riot squad. They carry guns and can be violent towards anyone opposing them. If you see the CRS on the street - go anywhere else, fast! They don't care if innocent bystanders get beaten up. If you are there, you are neither innocent nor a bystander.
The Gendarmarie Mobile are the Gendarmarie equivalent of the CRS. They seem to compete with the CRS to see how many people they can beat up.
Unlike British policemen, you don't argue with a French national Policeman. They can be more violent than British police are allowed to be, and don't usually face any punishment for hitting the wrong person.