Poison and Steel types are super effective against Fairy types.
Poison-types and steel-types
Fire type Pokemon are super effective against bug types
Almost any type can defeat a Fairy-type Pokemon, but its weaknesses are Poison and Steel. They are immune to Dragon and halves the damage to Bug, Fighting, and Dark-type moves.
Poison moves don't work very well on Rock, Ground, or Ghost Pokemon, and they don't work at all against Steel types... so if you get some of those type Pokemon to fight for you, you should have a good chance of winning. Try using a Pokemon with the move Earthquake, if you have it. Poison is strong against Grass types and Fairy types, so try not to use those types in the battle if you can help it.
Poison and Steel types are super effective against Fairy types.
Poison-types and steel-types
Fighting type is super effective against Normal type.
Dragon, Fairy and Ice-type moves. Fairy Pokémon are immune to Dragon.
Fighting types beat Normal, Ice, Rock, and Steel types.
dark types or ghost types
Electric types and Grass types.
Ground types and psychic types beat poison types.
I'm afraid there are no types of pokemon that can beat the Ghost type :P
Heres one for ya ghost types
Water, ground, and rock.
Fire Pokemon are weak against water types, rock types and ground types.