Burn
Fire types and Pokemon with Water Veil or Water Bubble cannot be burned. In Generation II, Fire types could be burned by Tri Attack.
Freeze
Ice types and Pokemon with Magma Armor cannot be frozen. In Generation II, Ice types could be frozen by Tri Attack.
Paralysis
In Generation I, a Pokemon cannot be paralyzed with a move that is the same type as itself. From Generation VI onwards, Electric types cannot be paralyzed. Ground types cannot be paralyzed from moves such as Thunder Wave, but a few other means, like Glare, can be used instead.
Poison
Poison and Steel types, plus Pokemon with Immunity, cannot normally be poisoned. In Generation II, Steel types could be poisoned by Twineedle. Poison and Steel types can be poisoned by a Pokemon with Corrosion.
Sleep
No types are immune to sleep. However, certain moves can be blocked by certain types.
Bound
Starting in Generation VI, Ghost types can switch out or flee even if afflicted with a binding move. Note that they're not immune to binding moves.
Can't escape
Starting in Generation VI, Ghost type Pokemon can still switch out or flee even if supposedly inflicted with can't escape.
Leech Seed
Grass types cannot be inflicted with Leech Seed in most circumstances. However, if a Pokemon uses Baton Pass while under the effects of Leech Seed, and switches to a Grass type, the Grass type will be afflicted by Leech Seed.
Center of attention
Grass type Pokemon will ignore a Pokemon that becomes the center of attention if Rage Powder was used. This is because Grass types are immune to the effects of powder moves.
Magnetic levitation
Ground type moves can't hit Pokemon that levitate via Magnet Rise.
Grass types
Grass type Pokemon are immune to Effect Spore, Leech Seed and powder and spore moves. While this doesn't give them full immunity from status conditions, it blocks poison from Poison Powder, sleep from Sleep Powder and Spore, health sapping from Leech Seed, paralysis from Stun Spore, and Rage Powder's center of attention effect.
Dark types
Dark types are immune to any status move -- and therefore their effects -- used by a Pokemon with Prankster. This doesn't make them immune to any status conditions, however, merely a few different ways of causing those status conditions.
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