Oysters can remain alive for 7-10 days when stored properly.
 

Always keep your oysters cold, but not frozen. Freezing kills oysters. Live oysters should be stored between 34-45 degrees F.
 

Direct shell surface contact with dry ice or gel packs can kill them also; place a cloth or other insulating surface between the live oyster and frozen surfaces if prolonged contact is expected.
 

Place the oysters on a towel or in a bowl in the fridge with the cupped side on the bottom (so the flat side is on top).  Be sure to cover your oysters with a damp cloth to keep them from drying out.  Re-wet the cloth when needed.
 

Never store shellfish in airtight containers, oysters can die from lack of oxygen
(NO Tupperware, ziplock, or vacuum sealing)
 

You can also store in a cooler covered with ice, but be sure to drain the cooler as the ice melts.  Never store shellfish in standing water!  Oysters should never be submerged in fresh water, sea water, or melted ice.  Keep your cooler as shaded as possible.
 

Oysters open their shells slightly to breathe - this is normal and good health. Never eat an oyster that is open or gaping! They should close when tapped. Discard any oysters whose shells do not close tight when tapped. If you hear a hollow sound when tapped, the oyster has lost its liquor and should be discarded.
 

Double check with a smell, the oyster should smell fresh and like the ocean, but not fishy or bad. Do not eat if the oyster smells bad.