My vehicle was struck by a tractor trailer and it was his fault, and I got the police form that shows the driver’s name. Whose insurance applies, the driver’s or the company he works for?

Both insurances apply, that is if the driver has his own policy covering his commercial activities, that policy applies coverage.  If he is working for a trucking company as their driver, moving their load, then the trucking company’s insurance will apply under federal regulations governing interstate trucking business.  Sometimes, the insurance covering the trailer, even if the trailer was not struck, can also provide additional insurance.