You need to use Soul Trap or a Soul Trap enchanted weapon on an enemy and kill them. When they die if there is a soul gem in your inventory that can hold their soul then the soul will be captured by it.
Soul Gems can be used to enchant equipment and recharge enchanted weapons.
"It's you Americans. There's something about nipples you hate. If this were Germany, we'd be romping around naked on the stage here." - Peter Molyneux
you can buy them either empty of filled with a soul, if theyre empty you need to either get a soul trap spell or enchant an item with soul trap. you then use the filled soul gems to enchant items at the arcane enchanter to enchant weapons/armor