Spells work when you learn the spell from a scroll and assign it to any of the quickslots on the right part of your screen at anytime once you have the rank and skillpoints required to learn that spell. Use the hotkey for that quickslot and then right click at any target(including companions) for most spells.

Each spell being cast will use up a certain amount of stamina and mana and will also have a cooldown time where you will not be able to cast any other spell for a certain amount of time depending on the spell being cast.

For offensive spells against hostile targets, you will also need to beat the target's spell resistance in order to cast the spell successfully. At low levels, spells may fail to work quite often. Assigning skillpoints to that school of magic, the skill Wisdom and having your alignment shift towards the alignment requirements of that school of magic will improve your chance at successfully casting that spell.

The spells Blade of Fire and Locust Swarm in the Magic of Miracles school of magic are good damaging spells you can get at some point during the first act. They are the most useful spells I have used all the way to the end of the game.