We all instinctively know what symmetry is, but describing it in words is harder than you might think. A picture of a butterfly is symmetric because you can reflect it in the line running down its centre without changing its appearance. A square is symmetric because it looks the same after it's been rotated around its centre by 90 degrees, or reflected in its vertical, horizontal or diagonal axes. A picture is symmetric under some transformation (such as a rotation or reflection) if that transformation leaves it unchanged. More generally, symmetry is immunity to change.
Like butterflies equations have symmetries too!
With this definition of symmetry things other than pictures can also be said to be symmetric. The mathematical expression gives the same result for as it does for so it's symmetric under the transformation that sends to
If you're making a time table for a school and have two teachers than can do exactly the same job and have the same availability, then the problem of making the time table is symmetric under the transformation of swapping these two teachers: if you have a solution, then swapping the two teachers also gives you a solution.
In physics, we know that the laws of motion are exactly the same in London as they are in New York, which means that the laws are symmetric under translations in space.
Thus, equations, problems and even laws of nature can all exhibit symmetry. In physics, that's a very important concept: physicists believe that the fundamental laws of nature should all exhibit certain symmetries, and they build those symmetries into their theories from the start, even if they haven't observed any direct evidence of them (find out more here). When it comes to real life problems, spotting symmetries can be very useful, as our teacher example above shows: if you know the symmetries of a problem, then one solution can immediately tell you what some of the other solutions are.
In ordinary life symmetry also plays an interesting role. Some have suggested that it's a prerequisite for beauty: we only find symmetric people beautiful because an asymmetry might be a sign of disease or other defect. On the other hand, a lack of symmetry can be attractive because it's different: we've all encountered people with a sexy squint or lopsided smile. Whatever your opinion of symmetry, it's definitely far more than just an abstract mathematical concept.
You can find out more about symmetry in these Plus articles.