Redox is short form for "reduction-oxidation." A redox reaction refers to any reaction that involves the transfer of electrons from one reactant to another, or a change in the degree electrons are shared. Taking an electron from a substance is called oxidation, while adding an electron to a substance is called reduction. The term reduction may seem counter-intuitive since you are adding an electron, but it refers to a reduction in the charge of the reactant. The reactant that gives up electrons is called the reducing agent. The reactant that takes electrons is called the oxidizing agent. If this seems mildly confusing, allow me to illustrate with a very simple example.
                     +          -
Na   +   Cl   -->  Na    +    Cl
Note for those with rusty chemistry: sodium has one electron in its outer shell, and chlorine requires one more electron to fill it's valence shell.

Here, sodium is oxidized and chlorine is reduced. The end result is table salt. In this example, an electron is actually transfered. In my last example I will use a common redox reaction in which electron sharing is altered. I give you burning methane.
 ------------ Oxidation -------
 V                            V
CH4      +   2 02     ->       CO2    +  Energy   +     2H2O
              ^                                        ^
              ------------------- Reduction ------------
Since oxygen is highly electronegative, the shared electrons in the carbon dioxide product are shifted away from the carbon. Since the covalent electrons in methane are evenly shared, the methane has been oxidized. Along the same vein, the covalent electrons are shared evenly in the oxygen molecule, but are shifted away from the hydrogen in the water molecule. Thus the oxygen molecule has been reduced.

Redox reactions often have oxygen as one of the reactants, such as burning, rusting and metabolism.