What was the first refrigerator?

Wow...two good answers.  I can't help but throw my hat in the ring, though.

It really depends on what your definition of a "refrigerator" is.  According to Wikipedia, a refrigerator is: A refrigerator (often called a "fridge" for short) is a cooling appliance comprising a thermally insulated compartment and a heat pump—chemical or mechanical means—to transfer heat from it to the external environment, cooling the contents to a temperature below ambient.

By this definition, we can't really call what the Chinese did a "refrigerator."  It's more of an "ice box."  While the box itself (or earth, as in the case of a cellar) would act as a "thermally insulated compartment" it would lack a heat pump.

Now, if you monkey the definition a little bit to say "A refrigerator is something that keeps things colder than the air around them," well, then, you could include "ice boxes" and that would probably be an example of the oldest fridge around.

Of course, like the first answerer said, it is hard to pinpoint these things.  If you go with the second definition, whose to say that some fella in Norway wasn't the first to keep his goat milk cool using hunks of ice...you can't really track it all that well.  The first definition, which includes a heat pump, is easier to research.

My money is on the Romans.  There are reports of Romans putting their food into clay jars, putting the jars into a shallow pan of water, and then having a slave sit there and fan the water (as the water evaporates, it has a cooling effect, thus cooling the food.)  In this case the clay pots (and, to a degree, the water itself) might qualify as "thermally insulating" and the human being acts as a heat pump (the fan acts to remove heat from around the jars.)  Though weird, that seems to count as actual refrigeration.

As for the type of fridge that we think of, well, that's another story.  The reason for disagreement on the subject is that people from around the world were working on similar ideas at the same time.  Just because a patent was filed in a certain country doesn't mean that someone hadn't invented a process earlier (and simply hadn't patented it yet.)  In addition, a lot of guys came up with different "parts" that would eventually make it into a commercial refrigerator, or came up with an idea for a fridge that they never made.  The first "home unit" didn't arrive until 1927 when GE took all these parts and managed to make them fit in a cabinet.

Of course, you could contrast this with air conditioning, which is essentially a refrigerator for human beings, but that is probably better left for another question!