In many ways, sugar made the British West Indies what they are today. Sugar was the major driver of the West Indian economy. Perhaps more importantly, it was because of sugar that African slaves were brought to the West Indies, where their descendants are the dominant ethnic group today.
In the mid-to-late 1600s, sugar became the major crop in the Caribbean. Needing workers to staff the sugar plantations, the British started to import large numbers of slaves from Africa. For example, this link tell us that the island of Barbados went from having only a few hundred slaves in 1640 to having 20,000 in 1650. By the end of the century, the slave population was up to 80,000.
This sugar economy led to a society in which there were huge numbers of black slaves ruled by a relatively small white population. At the apex of this society were the rich white plantation owners. This has meant, of course, that much of West Indian culture has been tremendously impacted by the cultures brought over by African slaves. Thus, the sugar economy had (and continues to have) a huge impact on West Indian culture.