What powers did colonial governments have in the 18th century?

Colonial governments had their own governors who operated with a great degree of independence from the home country's government.  They also elected their own legislative assemblies and made their own laws.  They could assess and collect their own taxes for purposes they decided upon within the colony itself.

If we take the American colonies as a primary example, the British home government for many years practiced a policy that Edmund Burke later would call "salutary neglect."  "Salutary neglect" meant that the British government "neglected" or took a mostly hands-off approach to the colonies, because that was "salutary" or beneficial to the success and profitability of the colonies.  Eventually, however, in the mid-18th century, the British government began trying to assert greater governmental control over the colonies, including matters of taxation.  This especially happened after the French and Indian War, in an effort to deal with war debt that the British government had incurred. The attempts by the British imperial government to exercise stronger control over the colonies eventually led to the Revolutionary War.