Plasma is one of the several phases of matter. A phase of matter, in thermodynamics, is a portion of a system that is uniform in composition. It can be thought of as relating to basic particles making up a piece of matter, and the nature and strength of the bonds between them.
For example, in a (crystalline) solid, there are strong intermolecular forces, holding the atoms together in a lattice formation, giving it definite volume and shape. In a liquid, these forces have deteriorated enough that it no longer has definite shape, but still has definite volume. In a gas, these forces are weak enough so that the atoms or molecules can move independently of each other, but they still remain atoms and molecules.
A plasma (or, more properly, an electromagnetic plasma) is a phase of matter that has enough energy for the electrons to separated from the nucleus. It consists of independently moving electrons and nuclei, and thus has some rather interesting properties, such as very good magnetic shielding. It can be found in places such as the center of the sun, and also, in small quantities, surrounding a lightening bolt.
Many people consider solid, liquid, gas and plasma to be the only four phases of matter. This is not true, as there exist many others, but they are generally more exotic with names like hadron gas or Bose-Einstein Condensate.
The plasma described above should not be confused with what biologists call plasma, which is a part of blood.
The Plasma Dictionary
William Zaientz, Physics Undergrad, Havard, Cambridge, MA