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The reactivity series, also called the activity series, is a list of metals that describe some of the ways in which they interact with each other.
Metals easily form positive ions (cations), which then bond with negative ions (anions) to form an aggregate neutral charge. This is one of the underlying focal points of electrochemistry, as well as the process of separating these ionic compounds and manipulating them in solutions. When separated, the positive metal cations and the negative nonmetal anions will sometimes be recombined with different materials to form new products. In a solution, all of the reactants can physically access each other, but compounds won't be formed by any and all of the available reactants; the reaction will favor the most "active" metal, i.e. the one that forms compounds more readily than any other.
In a reactivity series, metals are organized according to how easily they form compounds. Another way of thinking is that the higher the metal is in the activity series, the more metals below it will be displaced, or disfavored; a compound will prefer to be formed with a metal that is higher in the series.
This is useful in electrochemistry because it allows predictions to be made based on the reactants used. It also helps to describe some of the complexity of the electron shells, particularly the way in which the reactivity series doesn't always follow one's expectations depending upon the amount of information being considered.
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