"Put out the fire" is the most general and plain expression.
"To extinguish" means to bring to an end. It is more formal, and sounds a bit old fashioned.
As you can see here, "extinguish the fire" used to be much more common in the 19th century, but since 1880, "put out the fire" wins:
I think "extinguish" would likely have left the everyday English vocabulary completely, except that we have these things:
which we call "fire extinguishers". To some degree, "extinguishing a fire" might imply using one of these devices.
You can also "douse a fire" — to extinguish or put out a fire by dumping water on it.
To "fight a fire" is different — it's the process of getting a large blaze under control. Like this:
You don't fight the fire from a candle or a cookout — unless something has gone badly wrong.