“It is only slightly overstating the case to say that physics is the study of symmetry.”
P.W. Anderson (Nobel Prize in Physics 1977)
“More is Different“, Science, 177, 4047 (1972).

By symmetry, Anderson writes, “we mean the existence of different viewpoints from which the system appears the same.”

According to The Feynman Lectures on Physics,  Hermann Weyl gave this definition of symmetry: a thing is symmetrical if one can subject it to a certain operation and it appears exactly the same after the operation. For instance, if we look at a silhouette of a vase that is left-and-right symmetrical, then turn it 180∘ around the vertical axis, it looks the same.

Water drops, snowflakes, and crystals are examples of natural objects with symmetries. But symmetries in nature are not limited to objects. They exist in the basic laws which govern the physical world.

What does it mean that a physical law is symmetrical?

For example, let’s look at symmetry under translation in space and in time.
Experiments are routinely done at different labs around the world. Symmetry under translation in space means that the outcome of an experiment is independent of where it is done as long as all the relevant conditions are the same.

Experiments are also conducted at different times. Symmetry under translation in time means that their results are the same (assuming that the measurements do not depend on seasons and such). In fact, “different times” can be billions of years apart.

For each of the rules of symmetry there is a corresponding conservation law:

• Symmetry under translation in space means that momentum is conserved.
• Symmetry under translation in time means that energy is conserved.

For other important examples see The Feynman Lectures on Physics.

Quite surprisingly, the connection between symmetries in space and time and the fundamental conservation laws were not known during the nineteenth century, nor at the beginning of the twentieth. In a paper “E. Noether’s Discovery of the Deep Connection Between Symmetries and Conservation Laws”, Nina Byers follows the history of the discovery. Here is a shortened (and non-mathematical) version.

In 1915, Emmy Noether was invited to join the team of mathematicians assembled in Göttingen by David Hilbert. Shortly after she arrived, Albert Einstein gave six lectures in Göttingen, on the general theory of relativity. At that time the theory was not yet finished. However, the basic ideas were clear and his audience found them compelling. Although the general theory of relativity was completed in 1915, there remained unresolved problems. One of them was the principle of local energy conservation. In the general theory, energy is not conserved locally as it is in classical field theories – Newtonian gravity, electromagnetism, hydrodynamics, etc..

Hilbert asked Noether to look into the problem. Consequently, she began to study relativity theory. Out of that study came two papers which Hermann Weyl characterized as giving “the genuine and universal mathematical formulation of two of the most significant aspects of general relativity theory.”
Noether’s first theorem, which she proved in 1915, but did not publish until 1918, not only solved the problem for general relativity, but also determined the conserved quantities for every system of physical laws that possesses some continuous symmetry.

Upon receiving her work, Einstein wrote to Hilbert:

“Yesterday I received from Miss Noether a very interesting paper on invariants. I’m impressed that such things can be understood in such a general way. The old guard at Göttingen should take some lessons from Miss Noether! She seems to know her stuff.”

To get a more colorful account about Neother’s years at the University of Göttingen, I recommend to read “Women in physics in Fermi’s time”, also by Nina Byers.
To get the idea how is was, here is an xkcd cartoon