Depends. If it just becomes mushy or freezes very little, probably no damage will be done. if it freezes solid, you will have severe internal engine damage. Cracked block, or cracked head is common.
It freezes.
Assuming you still have some form of coolant (water) in your radiator then the coolant will freeze when the weather is cold enough, coolant expands when it freezes, this may cause permanent damage to your engine, such as split pipe's, burst radiator, cracked engine block
If the cooling system just has straight water in it , there is no corrosion protection for the cooling system , water pump , head gaskets etc , the engine '"coolant " can freeze or boil easier , if the coolant system freezes and the core plugs in the engine block don't do there job , the engine block can crack etc
Water expands when it frieze's. Well your engine has areas in it for coolant to travel and cool your engine. If the coolant frieze's and has no room to expand in those designated areas it can crack and burst those parts. Picture it as an over filled water balloon that ruptures due to the pressure
It explodes and the apocalypse happens.... What do you think happens? it freezes
Check the freeze plugs. large round plugs on either side of the block. Those are there for if your coolant freezes, they will go instead of cracking the block.
The remains of the old coolant can cause the new coolant to fail. Using the wrong type of coolant can also damage the engine.
After coolant is added run vehicle to normal operating temperature and re-check coolant level
You drain the oil immediately. If you run the engine with coolant mixed with the oil you will do serious harm to the engine. If you continue to drive it in this condition you will destroy the engine.
it's freezes
Freeze plugs are designed to blow out if the engine coolant freezes and then expands ( they are a " safety " device to hopefully keep the engine block / cylinder heads from cracking ) Straight water freezes at 32 degrees Fahrenheit , but a 50 / 50 mix of ethylene glycol antifreeze and preferably distilled water freezes at - 35
It has a much lower freezing point than water. if the coolant freezes it expands cauing pressure damage or even leaks. It also contains anti corrosion chemicals to prevent rust inside the engine.