General:
* Family car
* Station wagon (if it's a wagon-style vehicle)
* Minivan (if it's a minivan)
* SUV (if it's a sport utility vehicle)
* The family ride
* Our wheels
* The chariot
* The trusty steed
More informal:
* The family hauler
* The people mover
* The kiddie taxi
More specific:
* Mom's car
* Dad's car
* The soccer mom mobile
The best choice will depend on the specific situation and tone you're aiming for.