Generally speaking a category C write-off is one where the cost to repair was more than the value of the car.
A Category D write off is where the cost of repair is more than 60% of the value of the car. Insurance companies tend to write them off above this figure to avoid nasty shocks.
There is also a relatively new category X for stolen cars that insurance companies have paid out on and are then later recovered.
Some insurance companies will not cover category C or D cars - I think it's because their value is lower by virtue of their history and it complicates things if they have to pay out on them. But there are plenty that do.
If you're buying a category Cespecially you want to be asking how come it was viable to repair contrary to the category, what the damage was, can you see photos to verify what that damage was, who repaired it etc.
The reason for the ID check is because it's tempting for somepeople to buy a write off, steal a similar car and swap the identities - you'd want to be happy that this hasn't happened.