One of the most beautiful and general principles in physics was discovered by Emmy Noether. Hers is a fascinating story by itself, but I am a physicist not a biographer, so here comes some science-y stuff. Noether's theorem says that for every symmetry in a system, there is an associated quantity that is conserved. To see what that means in practice, it is useful to look at some examples.
Space is symmetrical in that (in the absence of matter like electrons, protons and galaxies), one piece of space looks very much like another piece of space. If I do an experiment in one part of space, then slide it over to another part of space and perform the experiment again, the result will be the same. This symmetry leads to conservation of momentum. If the second piece of space is different from the first piece of space (for example because it has a planet in it) momentum will not be conserved as a particle moves from one piece of space into the second (it will hit the planet and its momentum will change).
Time is also symmetrical in that if I do an experiment at one time and then do it again in the same place at a later time, I'll get the same result. This symmetry leads to the conservation of energy.
Another symmetry that space has is rotational symmetry. If I do an experiment with the apparatus pointing one way, then reorient the apparatus and do the experiment again pointing in a different direction, you'll get the same result. This symmetry leads to the conservation of angular momentum. Near the surface of the earth there is a rotational asymmetry due to gravity (there is a "special" direction - down). This assymmetry causes a pendulum to change angular momentum as it swings backwards and forwards (if you do it in a symmetrical place, such as far away from any sources of gravity, it will go around and around in a circle - it will have constant angular momentum).
Most of the time, our universe acts the same as it would if it was "flipped" the way a mirror-image reflection is flipped. This "mirror image" symmetry leads to the conservation of a property called the "parity" of a fundamental particle. The mirror image version of a particle has the opposite parity. However, it seems that there are some occasions when parity isn't conserved - in these respects our universe acts differently to a hypothetical "mirror universe", identical to ours in every respect except left and right being swapped. The apparent symmetry turned out to be an asymmetry.