In chemistry, a counterion (sometimes written as "counter ion", pronounced as such) is the ion that accompanies an ionic species in order to maintain electric neutrality. In table salt (NaCl, also known as sodium chloride) the sodium ion (positively charged) is the counterion for the chloride ion (negatively charged) and vice versa.
A counterion will be more commonly referred to as an anion or a cation, depending on whether it is negatively or positively charged.  Thus, the counterion to an anion will be a cation, and vice versa.
In biochemistry, counterions are generally vaguely defined.  Depending on their charge, proteins are associated with a variety of smaller anions and cations.  In plant cells, the anion malate is often accumulated in the vacuole to decrease water potential and drive cell expansion. To maintain neutrality,  ions are often accumulated as the counterion.  Ion permeation through hydrophobic cell walls is mediated by ion transport channels.  Nucleic acids are anionic, the corresponding cations are often protonated polyamines.