In geometry, a figure is chiral if it is not identical to its mirror image, or more particularly can't be mapped to its mirror images by rotations and translations alone. Such objects then come in two forms, called enantiomorphs. Chirality is equivalent to lacking a plane of symmetry (line in 2-D).
Many familiar objects are chiral - for instance, a right glove and left glove are enantiomorphic, and so are the S and Z tetrominoes of the popular video game Tetris. Some molecules are chiral, for instance optical isomers, and their study is part of stereochemistry. The fundamental laws of physics may be chiral, as the weak charge is not invariant under a reflection unless particle are replaced by their antiparticles as well, and kaon decay appears to violate even that symmetry.
Chirality appears to be important in particle physics because the universe seems to be asymmetric as far as spin is concerned. Imagine a particle moving in the direction of one's thumb. The particle can be classified as left-handed if it is spinning in the direction of the fingers of the left-hand and right-handed if it is spinning in the direction of the fingers of the right-hand. Up to now, only left-handed neutrinos (and right-handed anti-neutrinos) have been observed. But this is explained by the difficulty of detecting right-handed particles at extremely small masses.