Electronegativity is a measure of the tendency to attract electrons. Generally, it is used in the context of describing one species of atom's (element's) attraction of electrons in a chemical bond relative to another species. A higher electronegativity number indicating a greater tendency for attraction.
Thus, in an electrically neutral molecule consisting of two species of atoms bonded by sharing electrons, the electronegativity of an atom of one species measures its ability to attract the electrons it shares with the atom of the other species, the larger the difference in electronegativity of the two atom species the more time the shared electrons spend closer to the more electronegative atom and the more uneven the internal distribution of electrical charge in the molecule, thereby polarizing the molecule, something like a magnet with its north and south poles, the molecule as a whole remaining electrically neutral.
The Pauling scale (named after Nobel Prize winning Chemist Linus Pauling) is the first proposed and most commonly used measure of electronegativity. In this scale, Fluorine (the most electronegative element) is assigned a value of 4.0, and francium (the least electronegative) a value of 0.7.
Electronegativity is not strictly an atomic property however. The Miedema electronegativity scale is derived from the work functions of metal alloys and intermetallic compounds, and is thought to be more appropriate for applications involving layers of materials, like the Schottky diode, than the electronegativity scale of Pauling based upon covalent bonds in small molecules.  The Miedema scale is related to Pauling's scale approximately as XMied = 1.93XPaul + 0.87.
The various methods of "measuring" electronegativity actually indicate behavior of atoms in molecules. The equivalent property of a "free atom" is termed electron affinity. It has been observed that the electronegativity of elements can vary with environment. There is also a theoretical "inverse" of electronegativity, electropositivity, which is a measure of an atomic species' tendency to give up electrons to a chemical bond.
The practical effects of electronegativity can be seen in all life on Earth. The transfer of electrons between carbon (C) and oxygen (O) allows the storage and release of energy transmitted to Earth from the Sun.
The process of photosynthesis changes complex molecules from a lower to a higher potential energy (electrons are drawn away from O toward C). This requires a higher potential energy because Oxygen tends to attract the electrons more strongly than Carbon due to the relative electronegativities...thus, work must be done to effect this change and form a carbohydrate.)
Cellular respiration's net effect is to transfer electrons back down to a lower (C to O) potential, thus releasing energy for use in the cell. This phenomenon can be seen in other forms of chemical oxidation as well.
One note here, in the photosynthesis described above, Oxygen atoms can be said to undergo "Oxidation" in the sense that their electrons are drawn farther away from their nuclei. This is more a confusing semantic artifact of the definition of "oxidation" than a question of actual importance in Chemistry. The bottom line: In photosynthesis, oxygen is oxidized and Carbon is reduced. When "burning" the resulting carbohydrate, the reverse is true.
The relative electronegativity of two atoms also indicates important characteristics of the bond that forms between them. Bonds are often characterized as covalent when the electronegativities are basically equal, polar when there is some difference in electronegativities, and ionic when there is a greater difference. These characterizations are typically considered oversimplified generalizations by more advanced scholars however, and a more complete explanation of observed behavior of these different types of bonds has been attempted by applying Coulomb's law and principles of Quantum chemistry.
- A.R. Miedema (July,1973). "The electronegativity parameter for transition metals: Heat of formation and charge transfer in alloys". Journal of the less common metals vol. 32 (No. 1): pp 117-136. DOI:10.1016/0022-5088(73)90078-7. Research Blogging.
- W. Mönch (2006). “Equation (2.16)”, Susanne Siebentritt, Uwe Rau, eds: Wide-gap chalcopyrites. Birkhäuser, p. 24. ISBN 3540244972.