In meaning, these three verbs seem to be nearly interchangeable; all three denote a certainty or guarantee being made. However, in practice, the best writers do distinguish among these terms, as follows:

“Assure” is used to refer to interaction between people:

The editor assured me that my conclusion was incorrect.

“Ensure” is used more broadly to mean “to make certain”:

The company uses monthly financial disclosures to ensure clients of its solvency.

“Insure” is favored in instances of guaranteeing life or property against risk:

To insure your home against floods, you must purchase extra insurance separate from your homeowner’s policy.