HL7v2 interfaces are the oldest and still the most common path for high-volume clinical data: admissions, discharges, transfers, orders, and results. Epic's own interface engine, Bridges, is what a health system's IT team uses to configure and run these feeds. If you've heard someone on the hospital side mention an 'interface build,' this is almost always what they mean.
FHIR-based APIs are the newer, more app-friendly path. Epic supports SMART on FHIR for launching apps against structured resources like Patient, Appointment, Observation, and DocumentReference, plus Bulk FHIR for pulling data at the population level rather than one patient at a time. This is the layer most new integration work gets built against today.
App Orchard is Epic's own developer program for getting a third-party app registered, tested against sandbox data, reviewed, and eventually installed in a live Epic instance. It sits on top of the FHIR APIs rather than replacing them, it's the business and approval process around using them commercially, not a separate technical interface.
A couple of Epic-specific names show up often in searches without being integration paths at all, Epic Cosmos is a research database built from de-identified data across Epic organizations, and modules like Beacon (oncology) and Cupid (cardiology) are clinical workflow tools, not connection points. Worth knowing so you don't chase the wrong term.