Spells are the magical feats, be they religious, natural, arcane or psionic. Spells are categorized into spiritual magic or hermetic magic (SPIRITUAL MAGIC, HERMETIC MAGIC). Spells in any one guild will all be either spiritual or hermetic, not both. Typing 'info' in the joining room to a guild shows which type of magic a guild follows.
Every spell of both hermetic or spiritual types require spell points to be cast, although a few require endurance points or hit points as well. If you do not have enough spell points, you cannot cast the spell. To see what spell points are required, check the information on the spell by typing 'help spell <name>'. For example, 'help spell magic missile'.
The better your knowledge of a spell, the better your chance of successfully casting it. To increase your knowledge in a spell, you need to study it in your guild's training hall. Read the guild book there for more information on seeing what spells are offered by your guild and how to study them.
Spells require that you cast them to have any effect (CAST). Most spells require a target ('cast magic missile at bob'), while a few don't need one ('cast shelter'). To determine what targetting is required for a spell, read its help.
Your proficiency in a spell is also affected by the bulk of your guild and of any armours you are wearing. Read (BULK) for more information.