It's Super Effective!

I'm pretty sure everyone has had the message "It doesn't affect the foe's Pokémon..." at least once while playing. A great part of competitive battling is getting 'super-effective' hits. This involves knowing what type - or combination of types - your target is, out of 18, and how other types match up.

First, type matchups that do no damage whatsoever.
Normal/Fighting -> Ghost
Ghost -> Normal
Ground -> Flying
Electric -> Ground
Poison -> Steel
Psychic -> Dark
Dragon -> Fairy

In the case of Normal/Fighting vs Ghost, the ability Scrappy and the move Foresight will allow successful hits to be made. Everything else can only be stopped if the target is holding a Ring Target.

While some matchups are really obvious (Water beats Fire), others are a little obscure. To help you remember them, I'll include short explanations.

Key: (> Super effective) (~Not very effective)

Bug > Dark - Bugs can see in the dark.
Bug > Psychic - A psychic can't read a swarm's mind to defend itself.
Fighting ~ Psychic - A fighter's unchanging techniques are easy to predict.
Fighting ~ Flying - Martial arts don't work on birds.
Fighting > Ice, Rock, Steel - A karate chop with enough skill to shatter all three.
Ghost > Ghost - A ghost knows a ghost's weakness.
Dragon > Dragon - Magically charged fire breath breaks through another dragon's magic hide.
Poison/Steel > Fairy - Our poisons are foreign to fairies, so they have no resistance. 'Cold Iron' in legend is toxic to fairies.

For a full list of type matchups, this can be useful.

If you clicked the link, you'll notice moves can have quarter or quadrupled effectiveness. The single-type chart won't show any, but this is how multiple types interact with each other:
Type effectiveness multiplies damage. This means if one type resists your attack (halves damage) and another is weak to it (doubles damage) the final damage will be neutral effectiveness.
This also means a Dark/Fighting type still takes no damage from Psychic, even though it would normally hit 2x on Fighting.
As mentioned before, two weaknesses to the same type makes the Pokémon take four times the normal damage. Two resistances lowers it to a quarter.

If you don't know a Pokémon's type, either look it up or experiment to see what's good against it! Of course, you'll want to know the majority of their types before you start battling competitively!

Whether you're playing competitively or in story mode, you should always go for a 4x effective move when possible! (unless you're a hardcore player who's playing with intentional handicaps)

-Pokéxplain