As with any kind of asset owned by the deceased person, how you deal with bank accounts depends on how the person owned them.
If the deceased person owned the account in their own name and did not designate a payable-on-death beneficiary, then the account will probably have to go through probate. If the total value of probate assets is small enough to qualify as a "small estate" under state law, however, the inheritors will be able to use either simplified probate procedures or an affidavit to claim the money. Meanwhile, safeguard the money by transferring it to the estate bank account that you'll open.
These are the easy ones. The money is not part of the deceased person's probate estate, so you, as executor, don't have any authority over it.
The beneficiary named by the deceased person can simply claim the money by going to the bank with a death certificate and identification. The bank should have the document in which the account owner designated the POD beneficiary.
If the deceased person owned an account jointly with someone else, in most cases the surviving co-owner is automatically the account's owner. The account does not need to go through probate to be transferred to the survivor.
There can be exceptions to this general rule, however. Most accounts—but not all—that are held in the names of two people carry with them what's called the "right of survivorship." In other words, after one co-owner dies, the surviving owner automatically becomes the sole owner of the funds. (It usually works in a similar way with retirement accounts as well.)
Sometimes it's very clear that the account has the right of survivorship—for example, if the account is titled in the name of "Rachel Moore and William Moore, JTWROS." (The abbreviation stands for "joint tenants with right of survivorship.")
If the account registration document at the bank simply lists two names, and doesn't mention joint tenancy or the right of survivorship, it's probably still a joint tenancy account, but it might not be. Texas, for example, has a strict requirement that to create a joint tenancy account, the account owners must sign a separate agreement, in addition to the bank's registration card, creating the joint tenancy.
If two people—a married couple, for example—open a joint account together, no one is going to dispute that when one of them dies, the survivor owns the funds in the account. The situation may be different, however, when an older person adds someone else's name to their existing bank account.
Often this is done to avoid probate at the original owner's death. Sometimes, however, the second name is added only for convenience—that is, so the other person can write checks on the account, helping out the original owner. Or the arrangement is intended to give the second person easy access to the funds after the original owner's death, so that the funds can be used for the funeral or other expenses.
Legally, however, the person whose name was added to the account becomes the outright owner of the funds when the original account owner dies. Unless there's something in writing, there's no way for anyone to enforce the terms of whatever understanding was reached earlier, about how the money would be used. The new owner is free to spend it on whatever he or she chooses. Family members who are sure that the deceased person wanted a different result are unlikely to be successful if they go to court to try to get the money back from the surviving joint account owner.
Many people own valuable assets in a living trust, to avoid the need for probate proceedings later. If the deceased person held a bank account in trust, you'll be able to tell from the account statements; they will show that the account was owned by, for example, "Mara Chang, trustee of the Mara Chang Revocable Living Trust dated September 4, 2005." Like other trust assets, the account is under the control of the successor trustee, the person who takes over after the original trustee's death. It will be the successor trustee's job to transfer the funds to whomever inherits them under the terms of the trust document.