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 that nearly fills the pericarp, but does not adhere to it. In many species, what we think of as the "seed" is actually an achene, a fruit containing the seed.
The most familiar achenes are those of the strawberry, where the "seeds" are the achenes (technically the 'botanical' fruits), while what is eaten as the ('culinary') fruit is a so-called accessory fruit.
Fruits of sedges are sometimes considered achenes because they have a one-locule compound ovary. By the same definition, the common fruit type in the Family Asteraceae is also usually considered achene (some term the asteraceous achene cypsela, however). A sunflower "seed" in the husk is not really a seed, but an achene. The white-gray husks are the walls of the fruit.
A rose also produces achenes, which are nestled inside the rose hips (each rose hip, or the fruit, holds a few achenes).
A grain, a type of fruit closely resembling an achene, differs in that the pericarp is fused to the thin seed coat in the grain.
A utricle is like an achene, but it has a compound ovary, rather than a simple one. In addition, its fruit ovary becomes bladdery or corky.