There are some whose knowledge of the arcane mysteries of the universe is unsurpassed. Who, with a wave of their hand and the whispering of a mystical phrase, can unmake reality. Masters of the occult arts, it takes a lifetime for many to become truly accomplished wizards.
If you've been looking for ways to make your wizard stand out among the other spellcasters, the following list is a good place to start. As with other entries on this list, these tips are specifically meant for the Pathfinder RPG, but they may be useful for other games with a wizard class.
Tip #1: How did you learn?
Wizards learn magic with their minds, unlocking the intricate complexities of the arcane with nothing more than thought. However, it's important to know how your wizard learned the arcane arts in the first place. Did they attend a university, and learn under the tutelage of great sages? Were they apprenticed to a wizard? Were they a child of nobility, tutored by private teachers? Or did they get their hands on a grimoire, and learn magic on their own?
There's no right or wrong answer to this particular question, but it is worth thinking about. You could have been a work-a-day journeymen in the craft, just as you could have spent your four years at university. Or, perhaps, you won a drunken wizard's spellbook from him in a game of cards, and figured it was at least worth a read?
Tip #2: Why did you pick your specialization?
Every wizard has a school specialization (even those who choose not to have one fall under the umbrella of the Universalist wizard). When we choose our specialty, whether it's evocation, transmutation, necromancy, etc., we typically choose it for a mechanical reason. If you're planning on raising a lot of undead servants, you tend to go necromancer. If you want to be a party buff, you'll usually go transmuter. However, there should be more to this choice than just what spells it gives you bonuses to.
For example, did your wizard have a talent for enchantment, but chose to specialize in divination because of a family tradition? Was your wizard trained by the military, making evocation one of their primary focuses? Or does your wizard just have a talent for a particular school, even if they wish they didn't? In this case you might be an abjurer because abjuration spells just make sense to you, even though you'd like to be good at illusion. You can never get the illusions to go right, though, no matter how hard you practice.
Tip #3: What does your spellbook look like?
A spellbook is key to a wizard's power. It's where they record the rituals and components required for them to make magic happen. Without a spellbook, there is no way for a wizard to refresh their spells come morning (with the exception of read magic). So, since this item is so central to your wizard, ask what form it takes?
For example, is your spellbook a heavy, leather bound tome with a buckle on the front of it? Is it tattooed on you, allowing you to study your rituals with the aid of a mirror? Are your spells burned onto animals skins, and rolled up in your bag? Or do you have multiple spellbooks (fancy ones back at home, and dog-eared, hard-used ones as field journals)? Or is it more like a sketchbook, with a much-patched cover and a ring binder? Are the pages perforated for easy tearing?
You have a lot of options, and they're worth thinking through.
Tip #4: What do your spells and preparation look like?
While spells have universal descriptions, targets, components, and effects, that doesn't necessarily mean that everyone's spells look and sound exactly the same. As I said in What Do Your Verbal and Somatic Components Look Like?, you have a lot of leeway when it comes to your actual casting.
For example, do you cast evocation spells in orc, because the language is more forceful (perhaps giving a mechanical manifestation of the Maximize Spell feat)? Are your somatic gestures for abjuration akin to martial arts blocks, employing a warrior aesthetic to enhance an arcane art? Did you learn from a particular magical tradition, which has affected how your magic looks and feels? Or do you cast from a hodgepodge of different traditions, languages, and philosophies, making your own, unique style of casting?
Remember your spell preparation, as well. As I said in What Does Your Spell Preparation Look Like?, you aren't necessarily just sitting quietly and reading a book. Are you going through physical forms, as a monk would? Are you drawing shapes in the air, or on the ground, leaving the ritual complete but for a last word to activate it? Or do you prefer to sip your tea, and listen to the bard tune up while you refresh your necromantic energies?
Tip #5: The History of Your Familiar or Bonded Item
In addition to a spellbook, every wizard gains either a bonded item, or a familiar. While there are benefits to both in a mechanical sense, it's important to make sure the class feature you choose is an organic part of your character, and their history.
Say you chose a familiar. When did you get it? How long have you two been together? What is your relationship like? For example, if you have a raven, did you hatch it to ensure you had a close bond? Or did you acquire it as an adult? Did you teach it to speak, or did someone else do that? Is your raven old and curmudgeonly, or young and eager?
The same kind of questions need to be asked regarding your bonded item, if you go that route. For instance, does your class ring testify to your master of a certain school? Is your amulet a chain of office representing your position? Is that sword an officer's blade, given to all the wizards in the armed forces as an accessory of rank? Or did your grandmother give you her wand when you began study, helping you acquaint yourself with it. Does it take on a particular aura? Do you always put your arcane mark on it, showing your ownership?
These things aren't just class features... they're a part of your wizard. So incorporate them into your story, and you'll find they become a lot more than just a magic pet, or a free spell-per-day.