The paper regards banks as intermediaries between savers who prefer to deposit in liquid accounts and borrowers who prefer to take out long-maturity loans. Under ordinary circumstances, banks can provide a valuable service by channeling funds from many individual deposits into loans for borrowers. Individual depositors might not be able to make these loans themselves, since they know they may suddenly need immediate access to their funds, whereas the businesses' investments will only pay off in the future (moreover, by aggregating funds from many different depositors, banks help depositors save on the transactions costs they would have to pay in order to lend directly to businesses). Since banks provide a valuable service to both sides (providing the long-maturity loans businesses want and the liquid accounts depositors want), they can charge a higher interest rate on loans than they pay on deposits and thus profit from the difference.
Diamond and Dybvig's crucial point about how banking works is that under ordinary circumstances, savers' unpredictable needs for cash are unlikely to occur at the same time. Therefore, since depositors' needs reflect their individual circumstances, by accepting deposits from many different sources the bank expects only a small fraction of withdrawals in the short term, even though all depositors have the right to take their deposits back at any time. Thus a bank can make loans over a long horizon, while keeping only relatively small amounts of cash on hand to pay any depositors that wish to make withdrawals. (That is, because individual expenditure needs are largely uncorrelated, by the law of large numbers banks expect few withdrawals on any one day.)
This means that even healthy banks are potentially vulnerable to panics, usually called bank runs. If a depositor expects all other depositors to withdraw their funds, then it is irrelevant whether the banks' long term loans are likely to be profitable; the only rational response for the depositor is to rush to take her deposits out before the other depositors remove theirs. In other words, the Diamond-Dybvig model views bank runs as a type of self-fulfilling prophecy: each depositor's incentive to withdraw her funds depends on what she expects other depositors to do. If enough depositors expect other depositors to withdraw their funds, then they all have an incentive to rush to be the first in line to withdraw their funds.
In theoretical terms, the Diamond-Dybvig model provides an example of a game with more than one Nash equilibrium. If depositors expect most other depositors to withdraw only when they have real expenditure needs, then it is rational for all depositors to withdraw only when they have real expenditure needs. But if depositors expect most other depositors to rush quickly to close their accounts, then it is rational for all depositors to rush quickly to close their accounts. Of course, the first equilibrium is better than the other (in the sense of Pareto efficiency). If depositors withdraw only when they have real expenditure needs, they all benefit from holding their savings in a liquid, interest-bearing account. If instead everyone rushes to close their accounts, then they all lose the interest they could have earned, and some of them lose all their savings. Nonetheless, it is not obvious what any one depositor could do to prevent this mutual loss.
However, Diamond and Dybvig argue that unless the total amount of real expenditure needs per period is known with certainty, suspension of convertibility cannot be an optimal mechanism for preventing bank runs. Instead, they argue that a better way of preventing bank runs is deposit insurance backed by the government or central bank. Such insurance pays depositors all or part of their losses in the case of a bank run. If depositors know that they will get their money back even in case of a bank run, they have no reason to participate in a bank run.
Thus, sufficient deposit insurance can eliminate the possibility of bank runs. In principle, maintaining a deposit insurance program is unlikely to be very costly for the government: as long bank runs are prevented, deposit insurance will never actually need to be paid out. Bank runs became much rarer in the U.S. after the Federal Deposit Insurance Corporation was founded in the aftermath of the bank panics of the Great Depression. On the other hand, a deposit insurance scheme is likely to lead to moral hazard: by protecting depositors against bank failure, it makes depositors less careful in choosing where to deposit their money, and thus gives banks less incentive to lend carefully.