"Allspice, also known as Jamaica pepper, myrtle pepper, pimenta, or pimento, is the dried unripe berry of Pimenta dioica, a midcanopy tree native to the Greater Antilles, southern Mexico, and Central America. Wikipedia
Allspice got it name because it has the taste and aroma of cinnamon, nutmeg, and clove. You'll find it in a lot of Caribbean cuisine, most notably jerk seasoning. However, in Jamaica you'll often find jerk chicken is smoked using the wood from the Pimenta diolca tree instead of using the grounds from the berry.