Amalgam fillings are the metal fillings that dentist have placed in teeth for generations.  These fillings have a bunch of different metals in them.  These metals include tin copper and most infamous of all mercury. Which is why I refer to them as mercury fillings. Here is what the FDA says is in them.

The best way to remove metal (amalgam) fillings is to first isolate the tooth using a rubber damn.  This is a sheet of rubber which separates where the dentist is working from the rest of the patient.This technique makes sure that anything being drilled out of the metal mercury filling is not swallowed. 
Once the isolation of the tooth is completed a high speed suction is used so that the contaminated water and vapor produced while drilling is not ingested(swallowed)/ breathed in . The metal mercury filling is then drilled out in  chunks.This is done in the most minamally invasive way.
Depending how large the metal mercury filling is it can be replaced with new white (composite) filling or crown if it is too big to just fill.
Replacing metal mercury fillings can be done one at a time or a whole mouth and anywhere in between. It is up to the patient as to how long to stay at the dentist.