An associated state is a former colony that is now fully self-governing and could choose independence at any time, but prefers to remain in a comfortable association with the colonial power, letting them handle defence and foreign affairs. In some cases (as with many of the British West Indies) associated statehood was an interim stage towards independence; but in most of the remaining cases (Puerto Rico is a notable exception) the country may feel it is too small to run itself as fully independent.
The term Associated State was used for British external territories; United States possessions of comparable autonomy are called commonwealth. In the case of the Danish and Dutch territories, they are considered members of a partnership in the crown, so are something like the dominions that formerly composed the British Empire. The three British dependencies near the United Kingdom, viz the Isle of Man and the two Channel Islands, are not officially called Associated States but could probably be classed here.
The French have a much more uneasy and protective relationship with their colonies: they prefer to absorb them into Metropolitan France and quell the local separatist movements. But one place where natives outnumber settlers and have genuinely chosen free association is the Comoro island of Mayotte.