Major Caveat Here: I’ve never actually played with Wild Magic, and I’m also not familiar with every version of D&D, so this will be a more General than Specific answer.
Wild Magic is basically just a set of optional rules that make magic more unpredictable and random. As far as I can tell, most of the Wild Mage classes floating around are homebrewed. So be sure to consult with your DM first to see if they’re okay with this character concept and how they’d like to use the Wild Magic rules in their campaign.
Normally, when you cast a spell, you know what the effect of the spell is going to be. The spell description specifies things like target, duration, distance, damage, saving throws and whatnot.
The Wild Magic rules state that there are some situations where magic is less predictable, and casting a spell may result in a Wild Surge. A Wild Surge changes some aspect of the spell you cast. The change could be beneficial to you, or harmful to you, or weird but mechanically inconsequential. To determine the random effect of a Wild Surge, you roll a D100 on a table. (Here’s a Pathfinder Wild Surge Table, and here’s a more detailed and interesting one from 5E).
And that’s basically it. Wild Magic is just like normal magic, but sometimes something random happens. This is why you really need to talk to your DM before you decide if this is something you want to do. The DM is the one who decides when Wild Surges occur, and that’s something you’re going to want to know. As a Wild Mage, does a Wild Surge occur every time you cast a spell, or only in specific situations? Hash it out with them and decide if it’s something you’d like to try.
Wild Magic is cool because sometimes it will let you do something way more powerful than normal magic is capable of. But it’s just as likely to backfire and harm you. If that element of risk and randomness appeals to you, go for it! But if you’d rather be able to use your spells reliably, play a normal mage instead.