Plasma is one of many states of matter, but it is one of the most common four: solid, liquid, gas, plasma.
Plasma is matter where charged particles, nuclei and electrons, have been separated. This happens naturally in stars, where extreme heat and pressure frees the electrons around atoms. Lightning, flames, the ionosphere and auroras are also naturally occurring plasma, while neon signs, plasma TVs, rocket exhaust, and florescent bulbs are man-made, controlled plasmas. It is now thought that plasmas are the most common form of matter in the universe, where cooled plasmas reign supreme to fill in interstellar and intergalactic space with a sparse ion here and there.
Naturally occurring plasmas are further divided into two categories: terrestrial (lightning, neon signs) and astrophysical plasmas (stellar fusion, the heliospheric current sheet).
In solids, liquids, and gases, electrons play an important role in binding atoms together to form molecules. In this sense, electrons are like glue. Plasma by definition, separates nuclei and electrons, so molecules cannot form, and all parties are free to move around. Electromagnetism over large scales, binds plasma into non-molecular structures, and this is an area of intense research.
Plasmas are highly electrically conductive. Static sparks, plasma, are a common small scale release of ions from one object to another.
Plasma is not a strongly ordered state of matter (in contrast to a solid), nor is it strongly random (in contrast to a gas). Plasma is chaotic; it exhibits complex structure and behavior, especially if the matter involved can exist in the plasma state for a long period of time.
The heliosphere is the region of space around our star the Sun, and it extends far beyond the reach of the planets (or the icy planetismals such as recently demoted Pluto further out). In comparison with interstellar space, the heliosphere is filled with matter, flung out from the Sun as plasma on the solar winds (not wind in the sense that we know it on atmospheric Earth). This plasma, called the heliospheric current sheet, is loose, not very dense at all, but over the great breadth and length of the heliosphere, it is not uniform but rather has formed a roughly corrugated 3D spiral (spiral because the origin is the rotating magnetic field of the Sun).
New research has suggested that all of interstellar space is filled with different regions of matter density, separated discretely by walls of plasma. This cellular structure may be a remnant of the big bang just as the large scale structure of galactic clusters are.
On smaller scales, plasmas tend to form stringy short-duration stringy structures called magnetic ropes. Lightning is a familiar example, but this can also be found in the auroras and in man-made plasma lamps - those globe devices that every science center has that can be touched to attract glowing strings.