In 3.X wizards had spellbooks, while clerics had their entire spell list, but they both memorized/prepared spells every day from those lists, and had a certain number of slots for preparing spells in.
Now in 5e preparing spells and spell slots aren't the same thing. What's going on here? How does their spellcasting work now?
Things are different now. The Wizard and Sorcerer from prior editions have now combined into one class, called... the Wizard. Also, the Cleric picked up the same mechanics.
(Meanwhile, something mechanically new has emerged in 5e to take the name of 'Sorcerer', which has picked up some different stuff for its defining features, like a spell point mechanic.)
Credit to GMNoob, who taught me the basics.
If you've played 3.x or Pathfinder, you'll be familiar with this:
The spell process, from learning to preparation to casting, now offers the best of both prepared and spontaneous casting. I'll go through this level by level: what you know, what you have prepared, and what you can cast. Each level builds on the last.
There's a couple of features that grant you a lot of free spellcasting throughout the day. Both are described on the opening page of Chapter 10: Spellcasting, on page 78.
Cantrips: These are 0-level spells you always know and always have available to cast. You don't prepare these, and they don't use spell slots. The books say that "repeated practice has fixed the spell in the caster's mind". It seems that Wizards never even need to write them in a spellbook, since nothing ever mentions doing so, contrast to all the other rules for learning non-Cantrip spells.
Rituals: These let you cast a spell without using up a spell slot, or possibly without even having prepared it. It has to be marked as a (ritual) spell, it takes longer to cast, and it can't be empowered via casting through a higher level spell slot. The Ritual Casting feature of a class will describe further how they handle rituals.
Rituals will save a lot of spell slots in situations where you can sit down and take your time to cast a utility spell like Identify. For a Wizard, you can even leave those spells unprepared and save your prepared spells for other stuff.
For those who played Wizards and Clerics in 3.x and Pathfinder and might play one now in 5e, this means you now no longer have to fret about the proportion of spells: you won't run into a situation where you're saying drat, I wish I'd prepared that spell in just one more slot today.
For those who played Sorcerers in prior editions and might play the Wizard in 5e, your spell choices are no longer for life, as long as you can change and add to the contents of your spellbook, and you still get spontaneous casting. You don't have access to every spell you know all at once any longer, but if you're like me, you'll consider it worth the trade-off. (Or you could play 5e's version of the Sorcerer, which is entirely different in its own right.)
From the Basic Rules, p22 & p30:
The (class) table shows how many spell slots you have to cast your spells of 1st level and higher. To cast one of these spells, you must expend a slot of the spell’s level or higher. You regain all expended spell slots when you finish a long rest.
You prepare the list of (class) spells that are available for you to cast from (the cleric spell list / your spellbook). Choose a number of (class) spells from your spellbook equal to your (spellcasting ability score) modifier + your (class) level (minimum of one spell). The spells must be of a level for which you have spell slots.
For example, if you’re a 3rd-level (class), you have four 1st-level and two 2nd-level spell slots. With a (spellcasting ability score) of 16, your list of prepared spells can include six spells of 1st or 2nd level, in any combination, chosen from your spellbook. If you prepare the 1st-level spell X, you can cast it using a 1st-level or a 2nd-level slot. Casting the spell doesn’t remove it from your list of prepared spells.
You can change your list of prepared spells when you finish a long rest. Preparing a new list of (class) spells requires time spent (praying/studying): at least 1 minute per spell level for each spell on your list.
Essentially, 5e wizard/cleric spellcasting is sort of a combination of 3.X prepared & spontaneous spellcasting. If you are familiar with them, the spirit shamans from Complete Divine used an extremely similar system in 3.5, so they would be a good point of comparison. You prepare spells at the start of the day, but this merely determines which spells are available to you to cast that day. You have a certain number of spell slots for casting spells, but you don't choose which spell to use in them until you actually cast that spell. Note that, as in 3.X, wizards choose their spells to prepare from their spellbook, while clerics can choose any spells in the cleric spell list.
For example, a level 1 wizard with an Int of 16 can memorize four 1st level spells and has two 1st level spell slots. Let's say the wizard's spellbook contains burning hands, detect magic, mage armor, magic missile, shield, and sleep. The wizard chooses to prepare burning hands, mage armor, magic missile, and sleep that day. She can use her two spell slots to cast burning hands and mage armor, or sleep and magic missile, or magic missile twice, or any other combination of 2 spells from her prepared spells (duplicates allowed). She doesn't have to decide which of those spells she casts until she actually casts them & expends the spell slot. She couldn't cast shield, however, since she didn't prepare it that day.
Note that unlike 3.X you do not need to prepare/memorize a spell more than once in order to be able to cast it more than once in a given day.
A Brief Addendum on Higher-Level Slots
In 3.X you could prepare a spell in any slot equal to or greater than its level, which didn't offer any particular benefits, but let you cast, for example, a level 1 spell more times than you had level 1 spell slots. In 5e this is fairly similar: you can cast any prepared spell by expending a spell slot of its level or higher. In contrast to 3.X, however, most spells will automatically improve when cast using a spell slot of a higher level than needed. This helps make up for the fact that spells no longer scale with caster level. A magic missile cast from a level 1 spell slot is just as effective for a 1st level character as it is for a 20th level character; if the 20th level character wants their magic missile to be stronger then they need to cast it using a higher level spell slot.
Another Brief Addendum: Rituals
Both clerics and wizards can cast certain spells as rituals (spells that can be cast this way will say so in their description). Doing so increases the casting time to 10 minutes, but does not expend a spell slot the way casting it normally would. Note that wizards can cast any ritual spell in their spellbook this way, whether they prepared it or not, whereas clerics can only cast ritual spells that they have prepared.
It works much like a 3.X Sorcerer.
You have a limit of Character Level + Attribute Modifier preparation slots. These slots are not limited by spell level; each can hold a spell of any level the character can cast.
You have a separate list of spell slots that can be cast.
When you prepare spells, you make "fresh in your mind" the ChLv+AttMod spells. Until you next prepare, these are the available for use with the slots.
When you need to cast a spell, you pick any prepared spell. You also pick a spell slot that is at least as high a level as the spell. It uses the slot, but doesn't "unprepare" the spell.
If you want to cast it again, you can, as long as you have another spell slot that is big enough to cast that spell. You don't need to prepare it again.
Note that some spells can be cast as Rituals - these don't need to be "fresh" - they are cast using the spell book, and also do not take spell slots.
