Legally, a pet animal is considered tangible personal property, like your car, your furniture, your jewelry, etc., and upon your death, your pets would pass to your heirs or beneficiaries of your estate who are otherwise entitled to receive such property.  If you have a will or trust that does not gift your pet to a specific person, then the pet would pass to whoever receives of your tangible personal property.

If you have no will or trust, then your property would pass to your "heirs" under state statute, which usually means to:  (a) your surviving spouse; or (b) if you have no surviving spouse, your surviving children (and grandchildren by deceased children); or (c) if you have no surviving descendants, your surviving parents; or (d) if you have no surviving parents, your surviving brothers and sisters, etc.  Note, however, that this is a generalization, which applies in most situations and in most jurisdictions, but may not apply to you under particular circumstances.  For example, if you are married and have children from a previous marriage, the order of distribution under the applicable state statute may differ.

Of course, the above discussion is pertinent to the long-term fate of your pet – i.e., after probate proceedings have been initiated, debts and taxes have been paid, etc.  Very few states (such as Oregon) have laws dealing with the short-term care of your pet, and as such, making express arrangements for the immediate care of your pet is generally up to you.  At a minimum, you should have someone who will be timely notified and go to your house immediately in case of an emergency.  (See Pet Card, which should be carried on your person for emergency personnel.)