The essence of symmetry is that you can do something that, in general, should make a change, but it doesn't. Consider the small letter b. If you hold it up to a vertical mirror, you get a different letter, d. If you reflect b through a horizontal mirror, you get yet another letter, P. Thus b does not have any reflection symmetries. On the other hand, T is unchanged by a vertical mirror, so it is more symmetric. X is even more symmetric, since it is unchanged by either a vertical or a horizontal mirror.
A good example of symmetry in physics can be found in Newton's Laws. Students learn early on that these laws work just as well in any inertial frame of reference -- a frame of reference that is not rotating and is related to any other inertial frame by only a constant velocity. If you're a passenger in the front seat of a car driving at constant speed along a smooth, straight road, you can toss an apple to a friend in the back seat just exactly the way you would if the car were parked. Both this moving car and the parked car are in inertial frames of reference -- at least when we ignore the rotation of the earth, which is too slow to make a noticeable difference on the apple's trajectory.
In many cases, symmetry is present as a pattern. A good example are the three known "generations" of quarks and leptons. One generation consists of the Up and Down quarks (which combine to make protons and neutrons), the Electron, and the Electron Neutrino. Everyday matter consists mostly of these two quarks and the electron. Then another generation was discovered, piece by piece: the Charm and Strange quarks, the Muon, and the Muon Neutrino. Aside from the neutrinos, whose mass is not well known, these act just like more massive versions of the first generation. So when a Bottom quark, a Tauon, and a Tauon neutrino were discovered, physicists were convinced that there must be another quark to complete the set -- a Top quark. They were right.
This kind of thing has happened so often that physicists would definitely fall for this cartoon gag. The pattern must be completed!
Then there is the magnetic monopole. This is a hypothetical particle that would make magnetism more like electricity by being all "north pole" or all "south pole"; to keep the discussion simple, let's restrict our discussion to the "north pole" magnetic monopole. From whichever direction you approached this particle, you would see that you were approaching the north magnetic pole. This is not the case with familiar, real-world magnets; when you approach from one direction, you're getting closer to the north magnetic pole, but when you approach from the opposite direction, you're approaching the south magnetic pole.
There are two basic reasons why many physicists think that magnetic monopoles probably do exist, even if they have resisted all efforts to find one so far. (They may exist as possible particles, but be hard to produce in particle colliders and decay almost immediately when they are produced.)
1. Magnetic monopoles would make the equations of electrodynamics much more symmetric. Monopoles would act as sources for the magnetic field, similar to the way electric charges act as sources for electric fields and masses act as sources for gravitational fields. Also, just like a current of electric charges generates a magnetic field, and current of magnetic monopoles would generate an electric field.
2. The existence of magnetic monopoles would explain why charges are quantized, coming only in units of the charge on an electron.
Interestingly enough, "magnetic monopoles" of a sort have been observed in synthetic materials. This is less significant than it sounds. The "magnetic monopoles" occur only as quasiparticles in materials, and they apparently are not monopoles of what is called the B-field (the magnetic field of interest, as opposed to the H-field, which includes the effects of the material's magnetization). Also, materials can do strange things, bending the laws of physics beyond anything possible in a vacuum. "Electron" quasiparticles can have effective masses different from the mass of an electron; these effective masses can even be negative. Quasiparticles can have fractional electric charges, unlike real particles that can be observed in a vacuum.
Just a couple more points about the laws of classical electromagnetism. Maxwell's Equations were originally written as 12 separate equations involving the components of vectors. At the time vectors were a comparatively new mathematical idea, and Maxwell was not used to them. Oliver Heaviside rewrote them in vector form. This was not only more compact, it was more "beautiful", in that the vector formulation freed the equations from dependence on a specific coordinate system. Sometimes the geometry of the physical system is spherical (like a cloud of charge that depends only on distance from a point), sometimes it is cylindrical (like a long charged wire or a wire carrying current), and sometimes it is rectangular (like a rectangular waveguide). It is very helpful to be able to use coordinate systems with the same symmetry as the thing you're trying to describe, so it seems "beautiful" to be able to write Maxwell's Equations in a way that works for any coordinate system. In practice, though, after you choose the appropriate coordinate system you still have to expand the vector equations into sets of scalar equations for the components of the vectors in order to solve the problem. So you have a smaller set of equations to use, but they involve more complicated mathematical entities, and in order to actually use them you have to expand them back out again. If this seems complicated, then you are beginning to understand how the scientific idea of "beauty" is a subtle, acquired taste that is hard to explain.
That's not the end. In vector form, Maxwell's Equations require four equations, but they can be written in two equations if we use covariant (and contravariant) 4-vectors and rank-2 tensors. This formulation is more compact, but the complications are not really done away with, only hidden in more complicated mathematical entities. The advantage to this formulation is that it can be seen at a glance that these equations will satisfy Einstein's Special Theory of Relativity; the equations are now explicitly independent of any specific inertial frame of reference. Unfortunately -- you guessed it -- to use the equations you first have to choose an inertial frame of reference and unpack these into four equations, then choose a coordinate system and unpack them into (in principle) twelve equations. Beauty comes at a price.
You won't be surprised that these two equations can be combined into a single equation using even more hideously complicated mathematical entities by using something called Clifford Algebra. We spent so little time on that in grad school that I no longer remember what the point of that formulation might be. It is so far removed from being practical that almost no one seems to use it.
Beauty is an important guide in the development of theories, but beauty, as most adults have come sadly to know, can be deceptive. Philosophers say that in the final analysis, beauty, truth, and goodness are one; this may be true, but it is very bad advice to pretend they are identical in any but an eschatological "final analysis". The world is an astonishingly beautiful place, but we have no right to demand that it conform to our ideas of beauty.