Relic Run is a game type found in Heroes' Ascent. It is one of three game type options for The Hall of Heroes and the game type for Unholy Temples.
In a relic running match, a member of a team must carry a relic to either their Ghostly Hero or the central altar. The team who has captured the most relics wins. In the case of a tie, the person who scored last wins. If you kill an enemy's relic runner, he will drop the relic and your team can pick it up and return it to its spawn point.
The details on relic running in Hall of Heroes can be read here while details of relic running in Unholy Temples can be found here. The two maps have different mechanics but in general, teams prepare for them in the same way:
Two members of the team should be spared to run relics. Additionally, some component of the team is usually dedicated to snaring opposing teams. Lastly, the team should have some speed boost available to their runners to expedite running as the relic is a bundle and your movement speed is slowed while holding it. The favored skill in the current metagame for this is "Make Haste!" because, as a shout, it cannot be stripped, it provides a 33% increase in movement speed, and it can be placed on a single person's skill bar to provide a speed boost to two separate runners.