"Parity" is not a very easy term to define, especially since it means different things in different contexts. A very general definition would be "equality" or "equivalence". The word has many meanings in the sciences, depending on the field.
Parity is the quality of an integer that describes it as either even or odd. Note that because of what parity is, it must exist between 2 objects. Thus, you can say 1 and 3 have the same parity. 2 and 4 also have the same parity, but 1 and 2 do not. (Thus, if they are both odd or both even, they have parity.)
Stereoisomers are a class of important chemicals which have the same chemical bonds in different spatial arrangements. A subclass of stereoisomers are known as enantiomers (or optical isomers) - these are chemicals that do NOT have parity because they are, in effect, mirror images of each other and, thus, cannot be superimposed on each other.
Parity in biology is intimately related to enantiomers in large molecules, such as proteins. The two enantiomers of such functional molecules (usually known as the "L" and "R" varieties, for "left" and "right") have drastically different biological functions.
The even or odd quality of a string of bits in a binary code is called parity. This is most often used for error-checking processes, especially after transmission over a network. Usually, parity refers to the even or odd TOTAL of the bits in a string of code, not the uniformity of the bits (that is, they don't all have to be even or odd).
In physics, parity is an intrinsic property on the quantum level. It describes the symmetry of a wave function of that particle when it is reflected through the origin of the selected coordinate system. Unlike computer science, which describes parity with 1's and 0's, physics parity is described with +1's and -1's. Wolfgang Pauli's observation that no two electrons in an atom can have the same quantum numbers (thus describing the concept of "spin" is an extension of parity.
The term "parity invariance" has the simple, but somewhat unintuitive, definition that a process has the same chance of occuring if all the position, velocity, and acceleration vectors were reversed. Surprisingly, most processes ARE parity invariant. For example, it is true for both the strong nuclear force and the electromagnetic force. This property is, in part, what is responsible for us being able to reverse-calculate outcomes of radioactive decay. In 1956, it was discovered that the weak nuclear force is NOT parity invariant.
Well, what possible use is the knowledge that the weak nuclear force is not parity invariant? Yes, the Unified Field Theory, of course. An interesting side note: this reminds me of a riddle posed to me once in geek circles. You establish contact with an alien species that somehow manages to communicate to you from another universe. They ask you the simple question - what is "left" and what is "right"? (This assumes no visual communication, of course.)
Well, about the only way you can describe the concept of left and right is in physics terms. However, because parity is conserved in all fundamental forces except the weak nuclear force, the only way you can describe left and right are in terms of the weak nuclear force. HOWEVER...
Suppose the aliens now want to meet you. So you go to somewhere in space and see the alien, who we'll assume looks humanoid. You hold out your right hand to shake his hand, as you have told him is the human custom. The alien holds out his left hand instead... what do you do?
Remember that the alien won't learn the correct meaning of left and right if he is from an antimatter universe!
Yes, this "riddle" makes many assumptions, such as the fact that either you or the alien can go into an antimatter universe at all. But it's still an interesting way of illustrating what parity is and its importance.