What are these metal plates covering parts of PCBs called?

That is a shield and is used for many purposes. It can be to protect a sensitive area from EMI/RFI, or it can be to protect the rest of the board from EMI/RFI generated in that area or to help pass emissions testing.

If it is soldered down it can be problematic to remove as the large area acts as a heat sink preventing the solder form reflowing. Using a low temp heat removal system (using bismuth based solder) would work.

NOTE: this is not a "shield" as per Arduino usage - which is why the Arduino usage is a poor selection of terminology.

Existing answers are correct. It may be worth also noting that I've heard them referred to colloquially as "cans".

That is an RF shield, used to prevent emissions from the components below. Here is a Digikey link with results of a search