A Barth surface is kind of like a bumpy ball or a lumpy potato, but it's special because it's a surface that mathematicians study. Imagine taking a flat piece of paper and crumpling it up so that it has lots of hills and valleys all over it. That's kind of what a Barth surface looks like, except it's in three dimensions.
These surfaces were discovered by a mathematician named Wolf Barth, and they are interesting because they have a lot of symmetries. We use the word "symmetry" to describe things that are the same or balanced in some way. For example, if you have two halves of an apple that are exactly the same, that's symmetry.
Barth surfaces have a lot of symmetries that make them really pretty and interesting to mathematicians. They also show up in other areas of math, like geometry and algebra. Overall, Barth surfaces are just one example of how math can help us explore and understand the world around us.