The make of the unit is GE and below is a link to a picture of the label on the back. There aren't any other markings anywhere. I'm guessing this is an older model.
Here's a close-up of the ceiling of the refrigerator where the water seems to be leaking from.
I've read that the possible cause may be a clogged drain hole (with ice) and that leaving the unit unplugged for about four hours may do the trick. So I've unplugged it for now and am waiting. Thought I'd upload these pictures here to see if anyone can help further diagnose the problem.
Best Answer
From GE knowledgebase
I would check items 4 & 6.
For a clogged defrost drain GE says
Which isn't much help if you want to fix it yourself.
Here is an image of the cabinet assembly
And the unit parts (notice the drain tube is circled).
Images from Appliance Parts Pros.com
WARNING: If you are going to take the thing apart, make sure you unplug it first.