One of the things we observe in everyday life is that things have a distinct left and right. The simplest case is just the hands of a human: Obviously, the left hand and the right hand are different from each other. That is a very general thing in nature that things can be 'like a left hand' or 'like a right hand'. Of course, they do not need to be so. A ball has obviously no distinct left or right. But things can have. This fact is known in science as chirality, originating from a Greek word for hand.
Left and right are actually not that different. If you take a mirror, and look at a left hand in the mirror, it looks light a right hand. Such a process, which turns something behaving like a left hand into something like a right hand, is called a parity transformation in particle physics.
So far, so good, and some fancy names. Why should this matter? Indeed, it does matter quite a bit. In biology, molecules can also be chiral. And then it turns out that a certain handedness is nutritious for us, while the opposite handedness is at best useless and at worst toxic. Our body has a preference for a certain hand, it is chiral. The fact that the left-handed version of the molecule and the right-handed version of the molecule have different consequences implies that looking through the mirror is not always just a mirror image, but can be something entirely different. Parity is not just a change of perspective: The mirror image in this case is broken, and therefore one tends to say that parity, the property that something becomes just the mirror image without further changes, is broken.
So, what has this to do with particle physics? Well, also some elementary particles have a handedness. This handedness is an intrinsic property of such particles, such as a color for a billiard ball. This is especially important for the quarks and leptons of the standard model. Of each of them two exists: A left-handed one and a right-handed one.
When it comes to the strong interactions or to electromagnetism, this actually does not matter. For these two forces, both types of particles look exactly the same, and thus neither of these forces can actually distinguish between between left and right. These forces are also said to be parity invariant.
This changes when it comes to the weak interactions. The weak interactions are very special, and they distinguish between both types of particles. In fact, they are very extreme in this respect: The only act on the left-handed particles, but completely ignore the right-handed particles. It is said that the weak force is parity violating, or simply it is said that the weak interaction is chiral.
The consequences of this is quite profound, though not obvious. Take for example an atom with a nucleus which is unstable, and decays by emitting so-called beta radiation, i.e. electrons. If you suspend such an atom in a magnetic field, it turns out that the electrons emitted move in a preferential directions. This occurs, because the weak interactions are chiral. If they would not be, this would not happen. Nonetheless, this example shows that it requires something of sophistication to observe this.
Still, this chirality in the standard model is quite important. From a mathematical point of view, it is very restricting for the structure of the standard model. It has also quite important implications for each and every of our attempts to extend the standard model. Furthermore, in actual calculations it is quite a nuisance.
However, after all, we do not know why the weak interaction, but not the other two, are chiral. It is something we observe, and it is one of the bigger mysteries in particle physics. Therefore, looking for modifications of chiral properties is also a big chance to find something new. Since we have either perfect parity or not at all in the standard model, anything else would be new. Also, because we are so completely baffled by it, we think that whatever kind of observation is unexpected in context with a parity violation will very quickly leads us to a glimpse of whatever there is beyond the standard model.