An interesting question! The sun is like the biggest, hugest lightbulb you can imagine. Does a lightbulb have a dark side? No, a lightbulb shines equally in all directions. The sun is the same.
From Earth, we can only see the side of the sun that is facing us at any given time. Sometimes people refer to the side we can't see as the dark side of the sun. The sun rotates like planets do. It spins once approximately every 25 days, so over the course of this time period, all of the sun will be visible from Earth.
Scientists more commonly use the phrase "the far side of the sun" to describe the side of the sun we cannot see when it is facing away from us.
Related wonders:
How far away is the sun?
What are some other solar systems?
What is the dark side of the moon?
Image source: "Prominence Earth" by Lights In The Dark is licensed under CC BY-NC-SA 2.0.