If your cars alternator is putting out some voltage and working partially then yes the car will still run with the bad alternator.
However the alternator may be putting out just enough voltage for your car to run and not charge the battery though.
So if your alternator is not charging enough then your battery could be dead and you'd need a jump start of the battery to get started again.
If your alternator is not charging at all then the car will run with the bad alternator but only until the battery is depleted of it's charge.
Once the battery is discharged enough then the engine will likely shut off.
You can disconnect the positive battery cable while the car is running and if it still runs then the alternator is okay but if the engine dies then the alternator is bad and needs replaced.
An alternator should be putting out a minimum of 12 volts but ideally 14 volts is what the alternator in good shape should put out.