Assure means to encourage a person or restore confidence. If you can use the word “reassure” instead, then you can assure someone of something. The recipient, or object of this verb, is always a person.
Example: He did his best to assure her of his fidelity.
Ensure means to guarantee, to make sure that something is certain, or to confirm that it is true. The object of this verb is a concept, fact or situation, usually starting with “that.”
Examples: His job was to ensure that every customer was satisfied.
She proofread the numbers in the appendix tables to ensure accuracy.
Insure is only used in reference to the insurance business. If there is no insurance policy or insurance company involved, then don’t use “insure.”
Here is a great example from the University of Colorado at Boulder’s online style guide:
“I assured her that we would ensure that she was insured with our company.”