A Wizard by Any Other Hat
I'm probably not going out on too much of limb to say that when most of you think about wizards, you probably probably picture someone in long robes with a long white beard and a tall pointy hat. Something along the lines of Merlin (Sword in the Stone - Disney animation) or possibly Gandalf (Lord of the Rings - J.R.R. Tolkien).
In the world of RPGs, however, wizards can appear in numerous different varieties. In fantasy games such as Dungeons & Dragons, some of the possible wizard options might include, students of magic, wise wizards, mad mages, war casters, demi-liches, and even full liches such as the fabled Vecna. And even better than that, they aren't all old white human men with long beards, but rather they appear as any gender the player desires, any race that the player decides, AND any age that the player desires.
And if you happen to be stuck on the old white human male stereotype, here are a few pop-culture wizards that you might be familiar with who don't fit that stereotype:
Zatanna (DC Comics)
Jafar (Aladdin - Disney animation)
Ursula (The Little Mermaid - Disney animation)
John Constantine (DC Comics)
Jareth, the Goblin King (Labyrinth)
Glinda the Good Witch (The Wizard of Oz)
Lo Pan (Big Trouble in Little China)
Sarah Bailey (The Craft)
Ganandorf (The Legend of Zelda)
Wanda Maximoff - Scarlet Witch (X-Men / Avengers - Marvel Comics)
Now that we have unraveled the fact that anyone can become a wizard, the only real question remaining is why did your character choose to become a wizard (different from sorcerers and warlocks in D&D). The reasons for a character to become a wizard within an RPG are numerous, but let's see if we can think of a few options that you might consider while you are crafting your character's backstory.
What was it about magic that first drew your character toward wanting to become a wizard or witch?
Was your character introduced to the art of wizardry by their parents or was it by a beloved teacher (think Merlin teaching Arthur)?
Was becoming a wizard or witch merely a means to an end such as revenge, protection, knowledge, or ultimate power? You do know what becoming a Lich Queen does to your already perfect beauty, don't you?
Was the study of magic just something to do so that you didn't have to work on the farm milking aboleths?
Or maybe someone else discovered your talents and skills for magic and chose you as a student only to use your gifts and hard work to either protect the Republic or rule the galaxy (Star Wars Jedi or Sith)?
However your character came to be a wizard or witch, something to keep in mind (and to help keep them separate from D&D's sorcerers and warlocks), is that the skill of wizardry may not have come naturally to your character (sometimes this is based on character race), or it might have come very naturally. Either way, these characters will need plenty of study time for learning the spells, rituals, and alchemical knowledge that is required for practicing this powerful form of magic.
Unless your character is really good at teaching themselves how to do things, chances are they probably had a teacher of some sort along their journey and they have probably read a lot of spell books and scrolls to be able to learn the necessary skills and knowledge. By including how your character learned the necessary skills in your character's backstory, your Dungeon Master / Game Master may be able to pull out some of that information and incorporate it into your game's storyline to create even more reasons for your Wizard to be a part of this grand adventure.