Programmable Search Engine is a Google service for adding a customisable search box to your website, with results delivered by Google Search. You choose whether visitors search only your own site or also other sites on the web; available themes and a custom results page control the presentation. Results can be shown with advertising through AdSense or without ads. You create the search engine in a control panel and embed the search box with code. It is a hosted service; there is no self-hosted option.
Programmable Search Engine belongs to Google, part of Alphabet Inc., headquartered in Mountain View, California (US). The product page mentions no data processing agreement, GDPR reference or data residency, and Google's privacy policy says data may be processed on servers outside the user's country. Visitors' search queries therefore go to a US provider with the associated CLOUD Act exposure. European organisations should weigh this and may consider a site-search function that runs on their own infrastructure or at a European provider.