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A war of attrition is a war in which neither side has an exploitable strategic or tactical advantage, and the continuation of the war is just the slow bleeding by both sides of their strength. Thus, the war will be won by the side with greater reserves of personnel and war material, the loser eventually succumbing because they run out first.
A well known example of this is during World War I on the western front where both forces found themselves in static defensive positions in trenches that ran from the Swiss Alps to the English Channel. For years without any opportunity for maneuvers, the only way the commanders thought they could defeat the enemy was to continually attack each other head on and to grind the other down.
The Vietnam War has frequently been called a war of attrition, the American strategy being to wear down the enemy until he lost his "will to fight". Ultimately, this strategy would prove unsuccessful, perhaps due to the asymmetrical nature of the conflict.
If the sides are evenly matched or nearly so, the outcome of a War of Attrition may be a Pyrrhic victory.