As most people following physics research at some level or other will have noticed, physicists love symmetries. In fact, it can be and has been said that all of modern theoretical physics is based on a bunch of symmetry principles from which the rest follows.
While that may be a bit overly reductionist (experimental input plays an important part in the construction of a scientific theory after all), it is certainly true that symmetry considerations play a huge role in the building of our theories. But why is that so? The answer is that there are a number of mathematical theorems that link the existence (or absence) of certain symmetries in the mathematical formulation of a theory to physical features of the reality described by that theory: the laws of nature are constrained by symmetry.
Emmy Noether (1882-1935) (from Wikimedia Commons)
On of the first chapters in the story that led to this understanding of symmetry as the fundamental building block of physical theories was written by the woman in the picture above, Emmy Noether, who became a mathematician at a time when women in her native Germany still needed special permission to attend universities and were not allowed to teach students (which led to her having to teach under the name of her colleague David Hilbert).
Besides her many important contributions to algebra, Emmy Noether is most famous for the result known as Noether's theorem, which states that to every continuous symmetry of a physical theory there corresponds a conserved quantity, i.e. a physical quantity that does not change with time. This is a tremendously important result, since it allows us to derive conserved quantities from the mathematical form of our theories.
The proof of the theorem, while not hard to follow for an advanced undergraduate student of physics or mathematics, relies on the Lagrangian formalism of mechanics, which is possibly more detail than most readers would want, so I'll just give the gist of the idea behind it in non-mathematical term as far as that is possible for a mathematical result: A (classical, i.e. non-quantum) physical theory can be described in terms of a function called the action, which can be losely understood as assigning a number to any possible path through time that a system described by the theory could take. The path that a system will actually take out of all the ones it could take given its initial state is the one with the smallest action. If we now have a continuous transformation that we can apply to our system, we can construct a quantity whose rate of change with time is identical to the rate of change of the action under the continuous trnasformation. For a symmetry of the theory, the change of the action under a symmetry transformation is zero (that's why it is a symmetry), so the corresponding quantity does not change with time.
If you want a more mathematical sketch of the proof, look here, otherwise let us jump straight ahead to examples: The invariance of the laws of nature under time translation (i.e. the fact that they look exactly the same regardless of whether I put the zero of my timeline at the founding of Rome, the birth of Jesus, the hijra, or the Unix epoch) directly implies the conservation of energy, making perpetual motion machines impossible.
Invariance under spatial translations (i.e. that I can put my coordinate origin at the centre of the sun, at the centre of Sirius, or on the bridge of a hypothetical interstellar spaceship without noticing a difference to the form of the laws of the universe) implies the conservation of total momentum, so that to every action there has to be an equal and opposite reaction, and you simply can't have those wonderful zero-exhaust spaceship drives beloved of science-fiction authors. Invariance under rotations (i.e. the fact that there is no "up" and "down" in empty space) implies the conservation of total angular momentum.
Beyond these geometrically intuitive spacetime symmetries, there are many more abstract "internal" symmetries considered in particle physics that give rise to equally important conserved quantities. Electrical charge for example is conserved because the theory describing the interactions of charged particles with the electromagnetic field remains the same if we multiply the fields describing all charged particles by a complex phase eiφ.
It is generalisations of this kind of symmetry that form the basis of the Standard Model of elementary particle physics, and really most of modern theoretical physics.