A cars engine will still run without the serpentine belt.
However when the serpentine belt breaks or is not on the cars engine then the alternator will not charge the battery or produce voltage, the water pump will not work and you'll lose your power steering as well.
So you should not drive without a serpentine belt unless you need to get off the road and to somewhere safe.
I have had to drive a short distance such as 2 miles without a serpentine belt but only to get to a parts store that was close by.
Driving the car too far without the serpentine belt will make steering difficult as well as cause your engine to overheat and cause engine failure.
So you need to get the serpentine belt replaced before driving it too far.
Replacing a serpentine belt is usually pretty easy to do but some cars the serpentine belt is in a tight space and it takes some routing to get the serpentine belt installed properly and they can be confusing but you car should have a belt routing guide but if not it can usually be found online.
A new serpentine belt is between $20.00 to $100.00 depending on the make and model of the vehicle and an engine is more costly.