While many executor tasks can wait for a calmer and less stressful time, there are a few things you should do right away:
Inform close friends and relatives of the death. You can let other people know later, once things have settled down.
If the decedent was actively employed, and the employer might reasonably be counting on that person to show up the next day, it's nice to notify the employer. On the other hand, if the decedent was retired, or performed occasional contracting work, notifying an employer can wait.
Part of your duty as executor is to protect the estate. Unfortunately, it's not uncommon for criminals to use these situations as an opportunity to help themselves, or even for well-meaning friends or relatives to want to take something as a "keepsake". Things you should consider:
The executor often arranges the deceased person's funeral (see Funerals), although this is not a legal responsibility. As part of this process, you and your delegates will notify additional friends and family of the death.
If the decedent was a veteran, the executor should contact the VA as soon as possible, in order to potentially have them participate in and fund certain funeral activities and expenses (see Veterans). Your funeral director can coordinate this.
You will need several official death certificates for various government agencies, banks, etc. By far the easiest way to get these certificates is to request them via the funeral home or crematory at the time they file the original. You will have to pay $10-20 per certificate, and you will likely need at least 10 copies. If you later need additional copies, it can be quite painful to get them, so better to order too many than not enough.
If you do find yourself needing additional copies later on, see Task: Order death certificates for your state's specific instructions. Alternately, you can find an appropriate state authority yourself via the CDC, and you may even get more personalized service from the local county in which the death occurred.
Ideally, you already have a copy of the will, and know how to obtain the signed original. Get it, and keep it safe. See also Accessing Safe Deposit Boxes.
If there is no will, the deceased is considered to have died "intestate", and things will be a little more complicated. Each state has its own rules for determining who will inherit in such cases, and who should serve as executor. While you may want the help of a lawyer to get appointed as the executor, you may also want to use an Affidavit of Heirship to attest your relationship, and therefore your right to inherit ... and if the estate is small enough, bypass probate entirely.
See also First Month.