How will your bank accounts pass at your death? As with other assets you own, it depends on how you own the accounts during your life.
If you own an account in your own name, and don't designate a payable-on-death beneficiary (see below), then the account will probably have to go through probate before the money can be transferred to the people who inherit it.
If, however, the total value of your probate assets is small enough to qualify as a "small estate" under your state's law, then the people who inherit from you will have simpler, less expensive options. Depending on your state's law, they may be able to use a simplified probate procedure or simply prepare an affidavit (sworn statement) stating that they are entitled to the money, and present that to the bank. Not all states offer both options. To see what processes are available where you live, see Probate Shortcuts in Your State.
Probably the simplest way to leave a bank account to someone is to name that person (or more than one) as the "payable-on-death" or POD beneficiary. You can do it by filling out and submitting a form that the bank supplies. The money is not part of your probate estate (assets that can't be transferred without the probate court's approval), so it can be quickly and easily transferred to POD beneficiary.
After your death (and not before), the beneficiary can claim the money by going to the bank with a death certificate and identification. Your beneficiary designation form will be on file at the bank, so the bank will know that it has legal authority to hand over the funds.
If you own an account jointly with someone else, then after one of you dies, in most cases the surviving co-owner will automatically become the account's sole owner. The account will not need to go through probate before it can be transferred to the survivor.
Most bank accounts that are held in the names of two people carry with them what's called the "right of survivorship." This means that after one co-owner dies, the surviving owner automatically becomes the sole owner of all the funds. Sometimes it's very clear that the account has the right of survivorship—for example, an account titled in the name of "Roger and Theresa Flannery, Joint Tenants WROS." (The abbreviation stands for "with right of survivorship.")
If your account registration document at the bank simply lists your names, and doesn't mention joint tenancy or the right of survivorship, it might be a joint tenancy account, but it might not. If you're in doubt, check with the bank and make sure the right of survivorship is spelled out if that's what you want.
If you and your spouse open a joint bank account together, it's very unlikely that anyone would argue that the two of you didn't intend for the survivor to own the funds in the account. But if you have a solely owned account and add someone else as a co-owner, it may not be so clear what you want to happen to the funds in the account after your death.
Some people add another person's name to an account just for convenience—for example, perhaps you want your grown daughter to be able to write checks on the account, to help you out when you're busy, traveling, or not feeling well. Or you might want to give a family member easy access to the funds in an account after your death, with the understanding that the money will be used for your funeral expenses or some other purpose you've identified.
Legally, however, the person whose name you add to the account will become the outright owner of the funds after your death. Unless there's something in writing, there's no way to know or enforce the terms of any understanding the two of you reached about how the money would be used. The new owner is free to spend the money without any restrictions. If other relatives think you had something else in mind, they may be resentful or angry if the surviving owner uses the money for personal purposes instead of paying expenses or sharing the money with other family members.
If you want someone to have access to your funds only so they can use them on your behalf, there are better ways to do it. Consider giving a trusted person power of attorney (this gives them authority during your life), or leave a small bank account and instructions for its use after your death. Don't make someone a co-owner on an existing account unless you want them to inherit the money without any strings attached.
If you've set up a living trust to avoid probate proceedings after your death, you can hold a bank account in the name of the trust. After your death, when the person you chose to be your successor trustee takes over, the funds will be transferred to the beneficiary you named in your trust document. No probate will be necessary.
To transfer the account to your trust, tell the bank what you want to do. It may have some forms for you to fill out. Then the bank should adjust its records, and your account statements will show that the account is held in trust. For example, instead of getting statements addressed to Luanne O'Hara, you'll see statements to "Luanne O'Hara, trustee of the Luanne O'Hara Revocable Living Trust dated November 12, 2009."