Electronegativity is the tendency of some atoms to attract electrons more strongly than others. This is a simple yet profound concept, of huge importance in chemistry, which is tragically cursed with an intimidating eight-syllable name. I'm almost sure the unwieldiness of the term is the whole reason it gets left out of chemistry courses up to GCSE level even though plenty of things make far more sense once you start to get your head round it.
Oxidation
A great deal of chemistry relates to the way that electrons interact with atoms. Oxygen, for example, gets all its powers from the fact it's so attractive to electrons. This is why things burn in oxygen, why we're able to use oxygen to get energy from food, why oxygen corrodes metals and so on. In every case, it's because oxygen is an electron-hog: it's electronegative, to use the technical term. Grabbing on to electrons so tightly releases energy. Speaking of technical terms, when modern chemists talk about oxidation, what they really mean is that something is coming along and taking electrons. By the time chemists realised that lots of other things besides oxygen were also good at doing this, the name had already stuck. In any case, oxygen really is particularly good at hogging electrons; the only element that's more electronegative is fluorine.
Metals
By contrast, some elements are terrible at holding on to electrons. Their electronegativity is so low they count as electropositive. These elements are all metals, and the properties we associate with metals are all related to the way they hold onto their electrons rather loosely. They conduct electricity because some of their electrons aren't particularly attached to any one atom, flowing freely between them instead. This also allows metals to conduct heat rather well, high-speed electrons carrying heat energy rapidly through the metal. These free-flowing electrons hold the atoms of a metal together like wet glue, which is why metals are so malleable.
Metal atoms can break away from that structure by getting free of that electron glue. That means they have to form a positive ion, which usually means something else needs to come along and take at least one electron away. That could be a hydrogen ion from an acid, or it could be an electrode, for example. The most electropositive metals, like the alkali metals, only need the slightest nudge to lose an electron, which is why they're so extremely reactive. Ordinary water provides enough hydrogen ions for them to dump masses of electrons very quickly. The hydrogen escapes as gas, leaving behind a powerful alkaline solution, and the reaction produces large amounts of heat.
The fact that some elements are more attractive to electrons than others is the principle that batteries are based on. Put two materials in a conductive solution, with a wire allowing current to flow between them, and the more electropositive substance will send electrons down the wire and ions into the solution, completing a circuit.
Non-metals
Non-metals are elements that don't lose electrons easily. The more reactive ones are good at grabbing electrons from their surroundings, forming negative ions. Ionic bonding is when positive and negative ions come together to form crystals.
Non-metals can also bond together by sharing electrons, forming a covalent bond. If one of the atoms involved in a bond is more electronegative, the electrons are shared unevenly. The electronegative atom pulls the electron cloud towards itself, making the bond polar. Compounds with polar molecules behave more like ionic compounds than non-polar compounds do, because of that uneven distribution of charge: they tend to have higher melting and boiling points, because the opposite poles of nearby molecules attract each other. That's why water is liquid up to 100°C, even though it's just made of oxygen and hydrogen, each of which boils at temperatures hundreds of degrees lower than that on their own. Polar substances, including ionic compounds, tend to mix together, as do non-polar substances, but each tends not to mix with each other.
Trends in the Periodic Table
The reason that some atoms are more attractive to electrons than others has to do with how strong the positive charge is in their nucleus, how far from the nucleus their outside shell of electrons is, and how many other electrons are in the way. Electrons arrange themselves in layers around atoms, you see, and the top layer (the valence shell, or outside shell) is what does most of the chemistry. Going down a row in the periodic table means adding another layer of electrons, each of these shells being further out than the last. The further the electrons get from the nucleus, the weaker their attraction to it. That makes the metals more reactive, since they react by losing electrons, while the non-metals get less reactive. Lithium, the first of the alkali metals, fizzes gently when you put a lump of it in water; sodium, one row down, bubbles furiously, sometimes catching fire; potassium immediately bursts into purple flames; rubidium explodes; caesium is just nuts. The trend in the halogens (Group VII) goes the other way: fluorine is so good at poaching electrons that it reacts violently with almost anything, even at room temperature; chlorine is corrosive, and toxic to all life; bromine is still downright scary; whereas iodine is toxic enough to disinfect wounds, but not so toxic that doing so is a really terrible idea. Because all of these elements are so reactive, they are usually found as ions, which are much safer.
As you go across the periodic table, each element has one more proton than the last to draw the electrons in, and one more electron to balance it, but the new electrons are added to an existing shell, so they don't get any further out. In fact, thanks to the stronger charge on the nucleus, its attraction to electrons gets stronger and each electron shell gets even closer. Lithium, at the start of the second row, only has three protons, and the two electrons in its first shell get in the way of the attraction to the one on the outside. That makes lithium pretty electropositive. Near the other end of that row, fluorine has nine protons, so the force between that and the electrons in its second shell is far greater. Right at the end of the row is neon, wich ruins the pattern because, like the other noble gases, its outside shell is already full. It does hold on very firmly to the electrons it's already got, but it's got no space for any new ones.
So electronegativity, with its deceptively simple definition, draws together and helps explain redox reactions (reduction and oxidation), conductivity, the reactivity series, electrochemistry, acidity, ionic bonding, miscibility, periodicity and trends in the periodic table. Maybe it deserves those eight syllables after all.
Note that many authors prefer to discuss some of these ideas in terms of the related concepts of electrode potential and ionization energy, rather than tying everything together like this.