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Skill points allow you to learn skills and spells. Putting more than one skill point into a skill or spell increases your RANK in that particular skill.
Higher ranks increase effectiveness. This could mean more damage (i.e. kick, zap), longer duration (i.e. invisibility, sneaking), more health healed (i.e. heal, mage heal), reduced skill recovery time (i.e. bash), reduced stamina or mana costs (i.e. evade, parry), and so on.
Additionally, higher ranks are required for certain effects (i.e. more powerful shapechange targets, group version of spells like sanc or miracle, or the ability to use skills on mounted targets or while mounted, or additional effects such as knowing more from lore or identify.
For spells, increasing rank also increases the effective casting level of the spell. This is important to understand because most spell effects are based on casting level (not your character level!).
Here is a list of every skill and spell in the game
For more information, type help spec while logged into the game.