A "die" is just another term for "chip," where "chip" here refers to the actual unpackaged silicon. When you produce chips, you make wafers of them at a time. Then you slice and dice them into separate chips. The singular of "dice" is "die," so that's how the term evolved.
Today, "chip" can mean the unpackaged silicon (the die), or the fully-packaged integrated circuit, pins or bumps and all. A die only refers to the silicon piece of it.