For example, if you want to increase your sword skill, you will first need to purchase the swordsmanship skill from an NPC. Once the skill is learned, killing mobs with your sword equipped will increase your swordsmanship skill over time.
Most spells and skills can be purchased from NPCs, and they will teach you the basics of the particular skill or spell so you can start using it and training it. NPCs may ask you to help them perform certain quests before they are willing to teach you special spells and skills.