So I have a bunch of empty petty/common/grand soul gems. It lets me recharge my enchanted items with empty soul gems and I only have the first enchanting perk, which has nothing to do with recharging. What is the purpose of soul trap and what does Black gems do that others don't? Also, how come when I enchant items it lets me use whatever gem I want? I'm confused.

Soul Trap, is where you have a limit (60secs) I think to cast on an enemy if you kill the enemy with that time it will store that enemies soul into a Soul Gem. These can be used for Enchanting or recharging enchanted items.

They come in different shape and sizes and they hold different size souls.

A lesser soul gem will only hold a lesser soul I dont know like a mud crab for instance.
A Grand Soul gem will hold a soul of a giant for instance.
None of these can hold a Persons Soul, Thus Black Soul Gems were created to hold the soul of a person this is still only able to hold a grand soul or less.

If you use a grand soul gem to make an enchantment it will be more powerful than if you used a lesser soul gem, as the soul in the grand soul gem is larger.

Hope it helps.

erhm if you are recharging weapons with empty soul gems thats a bug (a delicious one), each sou, petty, lesser, normal, major,greater has a number behind it, the higher then umber the stronger the enchantment you make with it and them ore you can recharge a weapon with.
you need apropriate sized gems for the apropriate size souls to be harnessed, tho you can put a smaller soul in a larger gem.
and I honestly dont know what a black soul gem does in this game, in oblivion it was for a quest, dunno if it has any uses specific either than functioning as a normal soul gem.

As I said in my post, it lets you hold an NPC Soul. Obviously non-essential ones (ones that you can kill).

Soul trap lets you fill the gems with the appropriate level soul. You can either carry a wide variety of gems and use a soul trapping weapon (you'll ID it from the enchant that says something like "If target dies within X seconds, soul is trapped") or using the spell "Soul Trap".

For example:

Say you run across a cave bear in the wilderness, you get the bear to about 3% health, cast soul trap, bear dies, now the soul fills the "Common Soul Gem." You can now use that filled gem to recharge a magical item, or enchant a new one.

Black gems are the only type that can hold human souls.