Gems:
I don't care what you say, Shane, but I consider Gems to be separate from Earth.
Besides, gems are rare, so why not?
... I actually agree.

Oh well.
Fairy:
I consider Fairy to be different from Star. After all, there could be Dark Fairies, or Thunder Fairies, or even Data Fairies.
(Who knows what goes on in the Endanian world.)
Weirdness, that's for sure.
I suppose that'll work too.
Psychic:
I don't think Psychic is "magic". Therefore, I suggest this as well.
... You just said "psychic" for the purpose of
Pokémon, didn't you?
Moe:
Eh... heh heh. Sorry. 
I just think this ought to be an element 
In my opinion, this would be the most embarrassing way to defeat a monster
Kill monsters with
cuteness?!

...
WHAT.
Chemical:
This, I consider, would work well regarding the chemical elements (like chlorine, oxygen, gold, hydrogen, plutonium, arsenic, lawrencium, mendelevium, iodine and so on), and all things related to this element would deal damage of two elements (such as Chemical damage and Poison damage if it's caused by chlorine, or Chemical damage and Metal damage if it's caused by gold). This element would be the most difficult to implement, however, it's just a suggestion.
Well, this one I can understand, but it'll be confusing.
"Science" as an Element would be weird, too.
If you choose to use those elements, you can do with them as you like; abilities, weaknesses, strengths, you can choose. 
Okay.
...
So far, I agree with Gems and Fairy as elements here. I could agree with Chemical, provided I can get a better-sounding name for it.