So recently myself and a few friends have been playing the D&D 5e Starter Set at our local game store, as well as just starting a private game which included our wives, which was a blast.  When getting them set up for the game, we ran into the issue of a brand spanking new player grabbing, of course, the Wizard.

As I looked it over, there were some simple things and some things that, at least to some extent, weren't as clear.  I wanted to take a moment to clear up how spell casting works to help those who may have questions.

1. Spell Levels - Spells have levels ranging from 0-9.  The higher the level, the higher the number.  Note that spell level does *not* correspond to character level.

2.  Spell Slots - This can be something that confuses players but I'll try to make it easy.  The basic idea is that spells take a physical toll on the caster.  The less experienced [read: lower level] the caster is, the weaker their casting ability is, allowing them to only cast a certain amount of spells per day.  The higher the spells level, the more taxing it is to cast.  Makes sense, right?

Referencing the D&D 5 Basic Rules (found here) we'll use the level 1 Wizard for ease.  The Wizard leveling chart shows that a level 1 Wizard has two Level-1 slots (Level-0 spells are considered easy enough to not be taxing and as such don't have slots).  This simply means that the Wizard is able to cast two spells at Level-1 power each day (the time period between 8-hour rests).  As the Wizard levels up, he gets more spell slots, for example:

Level 1: Two Level-1 slots
Level 2: Three Level-1 slots
Level 3: Four Level-1 slots, Two Level-2 slots
Etc.

This shows the caster's ability to control more strong magic as they grow in experience.

3.  Spell Preparation - This, I think, can also cause a bit of confusion, mainly because its execution is accomplished mainly through role-play.  Essentially the caster can prepare an amount of spells per day equal to the caster's key attribute modifier + the caster's caster class level (this is to say, if your character decides to level up in another class, like Fighter, the levels they have in Fighter don't count).  So, to show it simply:

Amount of spells able to be prepared each day for a Cleric = Wisdom Modifier + Cleric Level
Amount of spells able to be prepared each day for a Wizard = Intelligence Modifier + Wizard Level

Also, a note worth clarifying, when preparing spells, you're preparing spell *types* not specific spells to be expended.  This means your caster is essentially making mental notes on how to cast X, Y and Z spells.  The spells you have prepared are what you choose from when picking spells to cast that fill your slots.

4.  Spell Casting - Casting the spell itself is easy; you simply read the description of the spell in the rules.

So, to break it down, this is what spell-related actions could look like in a day:
-Your character wakes up
-Your character thinks of spells he likes
-Your character is a level 1 Wizard with 16 Intelligence (giving him a +3 modifier), allowing him to prepare 4 spells he can choose to cast from later.
-Your character can only cast level 1 spells right now so he decides to prepare (make mental note of): Magic Missle, Shield, Mage Armor and Sleep.
-Your character gets in a fight and doesn't want to get hurt too bad, so he casts Mage Armor, a Level-1 spell, using up one of his two available Level-1 spell slots for the day.
-Your character then wants to shoot the bad guy, so he casts Magic Missle, another Level-1 spell, using up his last Level-1 spell slot for the day.
-He now must resort to using the level-0 spells he knows (which he can do as much as he wants because level-0 spells are easy), or get a solid rest in to restore his spell slots.

Easy peasy. :D

Edit: Due to an obscene amount of spam comments, I've disabled comments for this post.