A reducing agent (also called a reductant or reducer) is an element or compound that loses (or "donates") an electron to an electron recipient (oxidizing agent) in a redox chemical reaction.
A reducing agent is thus oxidized when it loses electrons in the redox reaction. Reducing agents "reduce" (or, are "oxidized" by) oxidizing agents. Oxidizers "oxidize" (that is, are reduced by) reducers.
Historically, reduction referred to the removal of oxygen from a compound, hence the name 'reduction'. The modern sense of donating electrons is a generalisation of this idea, acknowledging that other components can play a similar chemical role to oxygen.
In their pre-reaction states, reducers have extra electrons (that is, they are by themselves reduced) and oxidizers lack electrons (that is, they are by themselves oxidized). A reducing agent typically is in one of its lower possible oxidation states and is known as the electron donor. Examples of reducing agents include the earth metals, formic acid, oxalic acid, and sulfite compounds.
For example, consider the overall reaction for aerobic cellular respiration:

C6H12O6(s) + 6O2(g) → 6CO2(g) + 6H2O(l)The oxygen (O2) is being reduced, so it is the oxidizing agent. The glucose (C6H12O6) is being oxidized, so it is the reducing agent.
In organic chemistry, reduction usually refers to the addition of hydrogen to a molecule, though the aforementioned definition still applies. For example, benzene is reduced to cyclohexane in the presence of a platinum catalyst:

C6H6 + 3 H2 → C6H12