How do Soul Gems work?

Recently I enchanted a weapon with a fire damage enchantment. I found a better weapon and wanted to do the same thing with that weapon. However, this time, whenever I tried to select an enchantment, the game would say something like "charge not sufficient for this enchantment".

I figured that somehow the soul gems need to be charged with souls before I can use them. However, since then, I feel like I've killed at least a couple of dozen enemies, yet the message stays the same.

I feel like I haven't understood something about the soul gems. How do they work?

Solution 1:

In order to charge a soul gem, you need to use the Soul Trap spell: merely killing enemies isn't enough. Cast Soul Trap, then kill the enemy, and the gem will get charged.

If you have a weapon with the Soul Trap enchantment, that'll work as well.