While many mods can be run in the original engine, some make use of advanced features or cannot run within the limits of the original exe. You need then to identify which port, if any, is required for the mod.
You cannot normally run mods with the shareware or demo IWADs. The other Doom engine games, Heretic, Hexen, Strife and Chex Quest, each have only one relevant IWAD (although registered Strife, along with the map IWAD, also has a voices WAD).
Most Doom mods are made for Doom 2 but some were made for one of the other games in the series. The Doom series contains four different IWADs: Ultimate Doom, Doom 2, TNT: Evilution and Plutonia. You need to identify which IWAD is required to play the mod.