I know that most banks are FDIC insured and in those banks, the accounts are insured for up to $100,000. My question is this: Are banks required to be FDIC insured or is it optional? And if they are insured, are there any bank accounts that cannot be insured? I can't find information on these questions anywhere.

Not all banks are insured. A Bank would be prominently listed as a member of the FDIC.

And yes, some types of accounts are not insurable.