Symmetries and conservation laws

Symmetries and conservation laws lie at the heart of all physics. Newtonian mechanics can be summarized by conservation of energy, conservation of momentum, and $\vec{F}=m\vec{a}$, which describes how individual objects interact. But what is a symmetry, and where do conservation laws come from, anyway?
A symmetry can be viewed as a transformation that doesn't affect the way the world works. For instance, translational symmetry states that if you pick up the whole universe and move it one foot to the left, all the laws of physics remain the same. Another way of saying the same thing is that there is no such thing as absolute position; we can only measure the position of one object with respect to another. As a direct result of this symmetry, it can be (relatively) easily shown that linear momentum must be conserved.
Other similar symmetry-conservation correspondences include invariance under translation in time (start the whole universe five seconds later (kind of), and it works the same way) and the conservation of energy and invariance under rotation, which corresponds to the conservation of angular momentum.
These are all linked together by Noether's theorem, one of the most beautiful constructs in modern physics, which states that every symmetry corresponds to a conserved quantity.
Then there are the approximate symmetries. You find a lot of these in particle physics. They are symmetries that are mostly true. Often strong and electromagnetic interactions are symmetric, while weak interactions don't fully obey the symmetry. The most commonly mentioned ones are charge conjugation symmetry (C), parity transformation (P), and time reversal (T). All of these are discrete symmetries, which means they can be applied an integral number of times, in contrast to continuous symmetries like rotation and translation.
Charge conjugation is a transformation which when applied to a system causes all charges in the system  to invert. Just to confuse you, the charges that are reversed include not just electric charge but also things like strangeness, baryon number, and lepton number. Therefore under charge conjugation, every particle is transformed into its antiparticle. Note that applying a C transformation twice transforms the system back to its original state. In math speak, this means that the eigenvalues of charge transformation (which are called charge parity) are +1 and -1. The so-called eigenstates of C transformation are particles which are unchanged under a C transformation. These are their own antiparticles. All other particles are not eigenstates of C, and have antiparticles distinct from themselves. Other consequences of charge conjugation include the reversal of the directions of electric and magnetic fields, since these are caused by the presence and motion of electric charges, respectively.
I'm still a little confused by parity and time reversal symmetry, so I'll plan to write a bit about those sometime in the near future.