Like all European and many other countries, France operates on a GSM network. This means that if your current phone is GSM-compatible, it will probably work in France. If you are coming from North America or some parts of Asia you will probably have a CDMA phone, which will not work in France.
If you are in France and want to call someone living in France, dial a ten-digit phone number, always starting with '0'. If your want to call someone abroad, dial '00' + country code + phone number.
If you want to use you current GSM-compatible phone in France, you have two options:
Keep your current service provider from your home country: Although European law has capped the costs of roaming from home networks to more reasonable rates, this is still not a cheap solution as you pay for all of the calls you receive at international rates (for the leg of the call between your home country and France), in addition to being on more expensive tariffs for calls from France
Buy a new SIM card: This will give you a French phone number, so you can call on normal mobile rates and not pay anything for receiving calls. You need to check whether your phone is unlocked (some phones are specific to only one network). Even if your phone is locked, it is easy to unlock it (ask in a specialist telecom shop, unlocking your phone is in most cases free of charge.
There are four main providers in France, each with its own coverage and sales network: