Go to the bank and request the money if it was a jointly held account. If you're the other named account holder you can simply access the money as you would in a standard situation, since you have equal rights to the money. If you wish to have the deceased individual's name removed from the account, this is simple to do with a death certificate.
Bring the death certificate and proof of probate to the bank. If the individual left a will, the money in the bank account becomes part of the estate. It can be accessed only by the executor of the will as part of the probate process, after the court approves the will and the distribution of assets. If this is the situation and you're the executor of the estate, you will need to bring the required documents to prove your right to collect the money. This generally includes identification, court documents showing the will was probated and a death certificate.
Bring the death certificate and intestacy documents to the bank. If the person who passed away died intestate, or without a will, the court will make a determination on how the assets should be distributed according to the intestacy rules of the state. This will be written in a formal court decree. If this is the situation, bring the formal court decree to the bank declaring that you are the rightful heir to the money in the bank account. You will also need to bring identification to prove that you are the person listed in the decree as the rightful heir.
Bring proof that you are next of kin and a death certificate if the account was small and the statutory waiting period has passed. For certain small accounts, the proof that you are the next of kin and that the person has died is sufficient. However, a statutory waiting period--or a set amount of time as determined by the state--must pass before you can access the money this way. Banks also have different requirements for proving that you are the next of kin, so you'll need to contact your bank to find out the exact requirements. A birth certificate and identification are standard requirements.