If you have full coverage on your insurance and choose to file a claim under your own policy, your insurance will pay the cost to fix or replace your car, minus whatever your deductible happens to be. Deductibles are sometimes $250, $500, $1,000, or sometimes even higher. The way that works out is your insurance will pay the cost to fix or replace your car, minus that deductible amount, which you pay out of pocket. An example is if your car costs $2,000 to fix, but you have a $500 deductible. The insurance in this example would write you (or the body shop) a check for $1,500, leaving the balance for you to pay.
If you choose to file the claim under your own policy you will pay the deductible up front. This doesn’t change the fact that the at-fault driver is still on the hook for all the damages they caused, which are often paid back to your insurance by the other driver’s insurance. The problem that people often run into when going about fixing their car in this manner is that their insurance company may or may not pursue recovery for reimbursement of the deductible you paid. If they don’t get the deductible it is your responsibility to collect that from the other party.