What's the different between "require somebody to do something" and "require of somebody to do something"?
Or are they used interchangeably?
Thanks in advance.

the second construction is wrong.

'require of' is not used with infinitives. here's the way it's usually used:
NOUN is required of PERSON
so here are a couple of examples:
nothing will be required of you at the hearing.
what do you require of me?

the first construction probably needs no explanation. if it does, go ahead and post, and i (or another staff type) will explain it.

Hmm...how about this example?

The health insurance fee is required of all international students to insure that should any illness or injury befall them they are adequately insured to meet the cost of treatment.

that example is legitimate, but the infinitive isn't part of the construction involving 'required'.

note my example from before: 'nothing will be required of you at the hearing'. i wrote that as a complete sentence, but notice that its underlying structure is something like
nothing will be required of you // at the hearing
in this case, 'at the hearing' is a totally separate grammatical construction, not substantively involved with the 'required' part. (notice that, if you delete everything after the slashes, you still have a perfectly good sentence.)

your sentence can be parsed similarly:
the fee is required of all international students // to ensure that...
same deal with the slashes. if you write only the words before the slashes - 'the fee is required of all international students' - that's a perfectly complete sentence by itself. the rest of the words are auxiliary.

--

side point: you're confusing 'insure' and 'ensure'. this is not gmat-related (the gmat doesn't test homonyms), but it's still worth knowing.
insure = to take out an insurance policy on something
ensure = to make sure something happens
here's a sample sentence involving both:
the policy, which insures our house against fire and earthquake damage, ensures that all repair costs will be reimbursed in full.