A standard BigCommerce store runs on Stencil, BigCommerce's own theme engine, rendering pages server-side the traditional way. Going headless means detaching that frontend entirely: a separate application, a Next.js app, a mobile app, whatever a team builds, queries BigCommerce's GraphQL Storefront API for product data, manages cart state through it, and either hands off to BigCommerce's hosted checkout or builds a fully custom one against the Checkout API.
BigCommerce also maintains Catalyst, its own open-source Next.js framework purpose-built for headless BigCommerce storefronts, meant to shortcut a lot of the groundwork a team would otherwise write from scratch connecting a custom frontend to BigCommerce's APIs. It's a starting point, not a requirement; plenty of headless BigCommerce builds use a different stack entirely.