Almost no one holds paper certificates any more. Shares sit immobilised at the Depository Trust Company in the name of its nominee, and ownership moves by book entry between participant brokers. Your broker records that you own shares; the depository records that the broker does. That chain is called street-name-registration.
Its clearing subsidiary, the nscc, nets the day's trades so that only small residual obligations move, and guarantees settlement between members. DTCC also runs trade repositories used to meet derivatives reporting obligations.
The system is efficient and almost invisible until something goes wrong: corporate action processing, rehypothecation questions, and share recall during heavy short-selling all trace back to this layer.
Related: nscc, t-plus-one