The Federal Deposit Insurance Corporation was started during The Great Depression to prevent runs on banks and to provide for an orderly transition when banks fail. Before then, you had best pick your banks carefully, because if they went bust, you were just screwed. Now, if the bank fails, the government will cover you up to $250K per account type. The National Credit Union Administration provides the same coverage for credit unions.
Make Sure the Bank is FDIC Insured
It's basically stupid for you to have a single dollar in a bank account that isn't insured by either the FDIC or the NCUA. But you need to make sure you don't have more than $250K in a single type of account at a single bank. Type of account refers to the person's name on the account, not on whether it is a checking, savings, money market, or CD type account. You are insured for $250K for accounts in your name and $250K for accounts in your spouse's name. Joint accounts are divided between the two owners and added together with their “sole” accounts. So if you just want joint accounts, they're insured up to $500K per institution. (Although if you get divorced or widowed, remember that your deposit insurance limits just decreased dramatically.) Opening different account types (checking/savings etc), re-ordering the names, using and/or specifications, or using different social security numbers doesn't affect these limits. It's $250K, per person, per bank.
Retirement accounts at a bank, however, ARE a “separate category” according to the FDIC, and are eligible for another $250K in insurance. This could be useful if you want to hold CDs in the retirement account, but most of us aren't going to put a big chunk of retirement money into plain old cash at the bank and mutual funds, stocks, and bonds aren't covered by the FDIC, even if held in a bank's IRA account.
Revocable trusts, irrevocable trusts, and corporation/partnership accounts are also all separate categories, eligible for another $250K per person per category per bank.
It would be a nice problem to have to have to worry about these limits. That's a long way off for me, but if it isn't for you, no sense in not taking advantage of your “free” insurance. You can learn more at the FDIC website.