If the raid is part of Warlords of Draenor, Legion, BfA, or Shadowlands you can run the normal or heroic raids as much as you like; as Udiza said you’ll only get loot the first time you kill a boss each week for each difficulty level. So if you run one of those raids once on normal and once on heroic, you’ll get loot from each boss both times.
If the raid is from Wrath of the Lich King, Cataclysm or Mists of Pandaria then it gets more complicated as you can only kill each raid boss once per lockout period. So for example, if your regular raid group manages to kill the first 6 bosses in Icecrown Citadel on normal mode, you can’t go on to participate in a heroic Icecrown Citadel raid later that same week unless they’ve already killed those same 6 bosses. (The one exception to this is the Siege of Orgrimmar raid from Mists which follows the boss loot rule used by the newer raids.)
If the raid is from Vanilla or Burning Crusade, those raids only had one difficulty level so you can only run them once per lockout. In addition you are instance locked so you can’t join a different raid group even if they’ve already killed the same bosses that week.