• Unfrosted and Un-Cut Cake - Wrap these tightly in plastic wrap, top, sides, and bottoms, so the plastic is touching the sides of the cake (that is to say, don't just drape the plastic over the top). Second to this, go for a plastic bag with a "zip-lock" type seal. Store these wrapped cakes on the counter at room temperature, and they'll keep for about a week before starting to stale on you. Oil-based cakes tend to keep a day or two longer than butter-based ones.
To keep cakes for longer than a week, try freezing them.
• Frosted and Un-Cut Cake - The frosting actually acts like plastic wrap here, protecting the cake from air and moisture in the environment. A frosted cake can be kept at room temperature for 4-5 days. Cover it with a cake keeper or an overturned bowl to protect it from dust, pet hair, and other things in the air (plastic wrap isn't necessary and would smoosh up the lovely frosting). Parks says that she actually keeps her frosted cakes in the microwave!
• Cut Cake, Frosted or Unfrosted - Parks says to think of sliced cake as a living organism. As soon as you make a cut, moisture begins to escape and cause the cake stale more quickly. If you can, cover the sliced edges with more frosting to protect the cake from moisture loss. Otherwise, press a piece of plastic wrap directly onto the sliced side and make sure it sticks. Then proceed with covering and storing the cake as for the un-cut version. Cut cake keeps for a little less time, about 3-4 days.
• When to Refrigerate Cakes - Cakes both frosted and unfrosted, cut and un-cut, are perfectly fine at room temperature for several days. Refrigeration is only necessary if your apartment gets very hot during the day (As Parks says, "High moisture + high sugar + high temp = bacteria feeding frenzy") or if you're making a cake that won't be served for more than three days, like when baking cake ahead for a party or special occasion.
To refrigerate, wrap unfrosted cakes in plastic to protect it from absorbing any weird fridge smells and to protect it from drying out, and then unwrap it to warm up on the counter before serving. For frosted cakes, chill the cake uncovered for fifteen minutes to harden the icing, then wrap it in plastic wrap.
Stella, thank you for these very excellent cake storing tips!
• More from Stella Parks: BraveTart
Related: How to Frost a Layer Cake