Epic Arcs are replayable, extensive, branching mission arcs offering substantial rewards and great faction standing gains. The currently available Epic Arcs, as well as the Agents to talk to and the requirements to start each arc can be found within the "Agents & Missions" section of the The Agency.
For unavailable Epic Arcs, the standing requirements to have the agent issue the mission can be seen when hovering the mouse cursor over the red tab shown in the lower left corner:
Completing all missions of an Epic Arc, or cancelling a mission or mission offer during the Epic Arc will end the mission arc, incurring a 3 month waiting time before the Epic Arc can be restarted by talking to the respective agent again. When replaying an Epic Arc, pilots are not bound to prior choices and may explore other branches of the Epic Arc as they see fit.
As most other agent missions, rewards from Epic Arc missions may be shared between pilots in the same fleet. However, unlike Storyline Agent Missions granting unshareable faction standing gains, the faction standing gains rewarded by most epic arcs will also be shared among fleet members when the mission rewards are being shared.