An achene is a type of simple dry fruit produced by many species of flowering plants. Achenes are monocarpellate (formed from one carpel) and indehiscent (they do not open at maturity). Achenes contain a single seed. In many species, what we think of as the "seed" is actually an achene, a fruit containing the seed.
Typical achenes are the fruits of buttercup and buckwheat. The achenes of the strawberry are familiar, where the "seeds" are achenes. Technically, the strawberry is an aggregate fruit with an aggregate of achenes, and what is eaten is accessory tissue, so this is an aggregate accessory fruit. A rose also produces achenes. Each fruit, called a rose hip, holds a few achenes.
Image Above: Strawberry seeds, which are actually achenes.
Definition adapted from URL: http://en.wikipedia.org/wiki/Achene
Image from URL: http://en.wikipedia.org/wiki/File:Strawberry.achene.es.jpg