Which insurance is required by law
Compulsory motor liability insurance covers damage that you, as the driver, cause to somebody else - their vehicle, their property, their health. It is required for every vehicle entered in the Czech vehicle register that has not been formally taken out of service. The obligation follows the vehicle, not the driving: a car standing unused in front of the house still needs cover.
What it does not cover is damage to your own car. That is the job of comprehensive insurance, which is voluntary and worth considering for a newer or financed vehicle.
Registering a car and arranging cover
When a vehicle is registered in Czechia, proof of insurance is part of the process. In practice this means arranging the policy first and registering afterwards. You receive a green card as proof of cover - keep it in the car, it is what you show at a roadside check and abroad.
What decides the price
Two groups of inputs. The vehicle - type, engine capacity and power. And the driver - age, driving experience and the no-claims record, which has by far the largest effect on the final figure. A driver with a long claim-free history pays a fraction of what a newly licensed driver pays for the same car.
A record earned abroad is a common question. Some insurers accept it against confirmation from the previous insurer, others do not. Because the practice differs, comparing several insurers is worth real money here, not just a few crowns.
What to check before you sign
Look at the limits of indemnity. The legal minimum is the floor, not a recommendation - claims for serious injury reach into the millions, and the difference in premium between the minimum and a higher limit is usually small. Also check what assistance services are included and whether they cover the distances you actually drive.