I always thought certain types are weaker to others because in life the results are similar. Not because of cells unless we're talking about why poison Pokemon are weak to psychic types. I'll go through each type's weaknesses and explain why.
Bug: Fire, Flying, Rock
- Fire burns up bugs easily, Animals that fly (usually birds) eat bugs, rocks can easily squish a bug.
Dark: Fighting, Bug
- Fighting type Pokemon are usually seen as being "light" their moves and techniques are honorable while Dark is the concept of using sneaky moves to win. Good overcomes evil. Bugs are common at night it may be that bugs have an edge over dark types due to becoming used to the darkness/being too simple to comprehend the complexity of dark types and working around that.
Dragon: Dragon, Ice
- I think it plays on the idea that something can be so strong that almost only it can destroy itself. Which would explain why Dragon moves hurt Dragons. Ice could be that Dragons are usually thought of as reptiles as someone said before me. Being cold blooded means that ice could be especially harmful. The argument against this could be that fire should also be harmful, but I think the idea was that Dragons would have few weaknesses.
Electric: Ground
- Ground is a poor conductor.
Fighting: Flying, Psychic
I think the flying weakness may have to go with both grace and the fact that fighting types are grounded to some sort of platform. Fighting is not graceful, but flying is. Grace beats out sheer strength. As for psychics I think it has to do with the usual brains or brawn argument. In this case brains win.
Fire: Water, Rock, Ground
Water puts out fire. Rocks generally speaking could put out a fire. Dirt thrown over a fire will also put it out.
Flying: Electric, Ice, Rock
Electric moves could zap any flying animal/object out of the sky. Ice could freeze anything that's flying around. Projectile rocks would most likely throw a flying type out of the sky.
Ghost: Ghost, Dark
What else could defeat a ghost, but another ghost? Any living thing or object would just go through a ghost. They have a different make-up. Dark types could have the advantage because they are sneaky. They play unfairly and so they can hit ghosts. I believe that more so then dark types have the advantage because they are associated with darkness and ghosts typically come out at night.
Grass: Fire, Ice, Poison, Flying, Bug
Fire burns plants. Ice or a frost kills plants. Poison can be used to kill plants. Birds and such eat plants. Bugs eat/destroy plants.
Ground: Water, Grass, Ice
Water washes away and erodes the earth. Grass grows and sucks up nutrition through the earth. Ice can freeze into the earth causing cracks. Could also be on the idea of glaciers moving continents, essentially earth around.
Normal: Fighting
I see normal as being almost mammal like. Generally anything and everything can hurt you, but fighting seems to be the most likely and common of death. Fighting predators, fighting within your herd, fighting to survive.
Ice: Fire, Fighting, Rock Steel
Fire melts ice. Fighting moves break through ice same goes for projectile rocks. Steel could be that it withstands in ice/could break through ice.
Poison: Ground, Psychic
The earth absorbs impurities and eventually cleans them out. I've read that Psychic moves are strong over poison Pokemon because they manipulate their poison cells through psychic power into poisoning themselves.
Psychic: Bug, Ghost, Dark
Bugs are simple minded creatures perhaps the complexity of psychic power goes over their head. Therefore they can find a psychic Pokemon's weakness. Ghosts and psychics are connected in real life as being able to communicate with each other. Ghosts are usually draining and powerful to these psychic people and so that could have been relfected in the psychic type. Dark Pokemon again are emphasized on sneakiness. The can get past psychic power through their own devious methods.
Rock: Water, Grass, Fighting, Ground, Steel
Water washes away and erodes rocks. Plants can force their way through rocks, splitting them in half. Fighting moves can break rocks. The ground can crack and swallow up rocks. Steel can be forced through rocks.
Steel: Fire, Fighting, Ground
Fire melts steel. Fighting moves can break through steel. The earth can bend steel with immense pressure.
Water: Electric, Grass
Water conducts electricity. Grass absorbs water.