Performing an international data transfer requires either the receiving country to be deemed adequate or appropriate safeguards to be in place.  In general, the EU does not list the US as one of the countries that meets this requirement and therefore a data transfer to the US requires further safeguards.

The Privacy Shield was a certification for US companies which the EU recognised as providing adequate safeguards.  It only applied to the company that the data was being transferred to and then only for the specific service.

From a practical perspective relying upon the Privacy Shield was relatively simple, visiting the Privacy Shield website and ensuring that the company (and service) you are transferring to was listed.

The Privacy Shield has now been replaced in the EU by the Data Privacy Framework (dataprivacyframework.gov)