The Wizard is a pure spellcaster, harnessing arcane and elemental magic to do her bidding. Similar to the Sorcerer and Sorceress of previous Diablo games.
The word "wizard" is used derogatory for rogue Sorcerers and magi that are dangerous to the world at large in the world of Sanctuary.
The female Wizard character in Diablo III is from the island Xiansai, an island to the far north with a culture similar to Asia in our world. Abd al-Hazir writes about a 'wizard' wreaking havoc in Caldeum in his seventh entry of the Writings of Abd al-Hazir.
This female 19 year old wizard was sent to the Zann Esu mage clan, but later handed over to the powerful Vizjerei since she was a "rude and uncooperative student," that the smaller mage clan could not control. They hoped the harsh discipline at Vizjerei "would break her anarchic spirit." Yet even the Vizjerei instructors were unable to rein her in. She was continually being caught seeking out dangerous and forbidden magics without care for herself or anyone around her.
It was rumoured in Caldeum that she ventured into the infamous Bitter Depths below the Sanctum, but the truth is she was caught in the Ancient Repositories, where the most dangerous incantations are housed for the safety of the public.
Valthek, a great Vizjerei mage confronted her and she attacked him. She somehow relied on trickery and deceit or some sort of ancient incantations to render Valthek unconscious and then proceeded to flee the city immediately after the encounter. The extensive property damage was assumed to be the result of Valthek's magical prowess, not the young wizard.
Abd al-Hazir is concerned about the young, rebellious and inexperienced wizard wandering the world, using magics she does not understand. He mentioned that certain schools of magic were too dangerous and "those wiser than you or I determined long ago [to forbid] their practice." Primal forces including time warps include the secrets he suspects she gathered from the Ancient Repositories.
The above is the background of the female Wizard. Since it's specifically a woman, we don't know yet where the male version comes from, his background or motivations. We can assume he'll be every bit as arrogant and self-absorbed as the female Wizard, however.
The Wizard is an aggressive archetypal magic wielder founding his powers on the aggressive fierceness of the storm, the arcane and conjuration. Both the male and female wizards are rebellious, headstrong, impulsive, and brilliant. Too smart and arrogant to fit into the authority of the sorcerer schools of the mage clans. Always on the top of their skills and even smarter than their teachers, cocky and self assured.
Some fans have criticized the Wizard for just being the Sorceress with a new look. Jay Wilson commented on that in an interview with 1up.com in December 2008.
- I would respond, "Yeah, you're right; the Wizard is basically a reskinned Sorceress." What we couldn't do with the Sorceress very well was break into what I'd call the old-school pen-and-paper magic user. You know the old magic user who could do a variety of things, like conjure up animals out of midair or create clouds of fog and acid or control time or disintegrate things or use death spells. They had this wide variety of magic that they could use compared to the more traditional elementalist -- which is what the Sorceress is, meaning fire, ice, and lightning -- who was just more limited. What we really wanted to do is break into this area, while if we just did the Sorceress again, we'd be like, "OK, you have to do fire, ice, and lightning, but where does disintegrate fit? Where does slow time fit in?" So we decided [that we'd] just take the same class mechanics, and [that we'd] change the basic concept and name and just have a throwback to that old-school magic user to give ourselves a broader range of magic skills. But there was never a huge desire to go away from the basic gameplay of the Sorceress; there's a lot of repeated skills, and that's intentional.
One of the key changes to spell casters in Diablo III is the addition of +% spell damage modifiers. These sorts of modifiers were found on many skills in Diablo 2, and on a few uniques and runewords, but they were uncommon, and not necessary to kill effectively. As a result, casters in Diablo 2 were much less item-dependent than the combat classes, and found it easier to concentrate on Magic Find, resistances, hit points, and other non-damage bonuses.
This will not be the case in Diablo 3, where mages must add considerable +% spell damage from modifiers to stay ahead of the monster hit point curve. A high level Wizard or Witch Doctor in Diablo 3 will no more be able to kill quickly with no +spell damage than a Barbarian or Monk could succeed with a junk weapon. A quote from Flux's post-Blizzcon 2009 Wizard write up explains this in more detail:
- When viewing the listed damage on each spell, it’s important to keep in mind that those are not absolute numbers, in D3. They’re much like weapon damage; modified by your wizard’s attributes, your character level, by other skills, and also by your equipment. Lots of types of items in D3 carry modifiers that boost spell damage, usually by a percentage.
- I found numerous wands, staves, and even pieces of armor with values between +5-25% spell damage, and that was just in the early stages of the game. We know nothing about higher level equipment bonuses, and it’s entirely possible that the D3 team is projecting high level characters to have +100%, +200%, or who knows, +500%, spell damage. If so they’ll be adjusting the base spell damage accordingly, which might make it look very low to our untrained eyes.
- Here’s an example from early in the game, that was mentioned in the Witch Doctor skills report. The skill tree listed damage on my level 2 Skull of Flame spell was 6-9, yet with a level 12 Witch Doctor my Inventory screen showed 19-29 for that spell. I don’t think I had more than +50% spell damage from equipment, so assuming the display values aren’t wrong (in D2’s infamous LCS style), there are considerable bonuses added to damage from character attributes, which are then further boosted by the spell damage bonuses on your equipment.
- This is all part of the D3 team’s plan to make all of the attributes useful to all characters, and to make more types of modifiers useful/essential to spell casters. We don’t know enough yet to judge how well it’s working, but it should be a substantial change from how spell damage was calculated and scaled up (or not) in D2.
An off-screen still image of the Wizard's skill tree.
The wizard skills are arranged into to three distinctive skill tree categories:
All three trees have a combination of active and passive skills. Active skills are used in combat, while the passive skills boost or change the behaviour of the active skills.
The Arcane tree skills consists of many classical pen and paper RPGs, including time distortions, Disintegrate, and other magical forces that are not related to elements like Fire, Ice, Lightning. This tree is also good at enhancing or manipulating other spells known by the Wizard.
The Storm tree skills specialize in the power of the heavens, enabling the wizard to wield electrical energy and casting forth the fierce cold of a blizzard accompanied by the harsh freezing wind. The storm elements also protect and enhance the armour and the power of a wizard.
The Conjuring tree skills summons in protection or offence from other planes of existence. This is where the classic Hydra spell is located, as well as the powerful meteor. The tree can also be used to increase or modify other spells, similarly to the Arcane tree.
The Wizard was the third class revealed by Blizzard, debuting at BlizzCon 2008. It's in essence a re-make of the Sorceress, but with a slightly different set of skills.
In the GamesCom 2009 demo, we saw the male version of the in-game wizard the first time.
22nd of September 2009, Bashiok announced the Wizard will not be using traditional Mana, but a resource more specialised for the class. 
You can find pictures in the Diablo 3 screenshot and picture gallery: