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 is a certification for US companies which the EU recognises as providing adequate safeguards. It is important to note that it only applies to the company that the data is being transferred to and then only for the specific service.
From a practical perspective relying upon the Privacy Shield is relatively simple, visiting the Privacy Shield website and ensuring that the company (and service) you are transferring to is listed.
The Privacy Shield website can be found here: https://www.privacyshield.gov/list