It is crucial to understand that "size" when used with digital image files applies to two separate classes of measurement: area (measured in pixels along orthogonal axes of height and width) and storage requirement (in effect, volume; measured in bytes). The two measurements are related, but not in a way that can be of much use. The relationship is direct for file formats that do not compress image data (e.g.: TIFF as it is usually used) but is not direct for file formats that do compress image data (e.g.: JGP). There is no useful way to compare the relationship from one file format to another.
You might envision this as surface area and volume of a physical object. A ball might have a surface area of 1 and a volume of 1. If you roll the ball out so it is more like a hot dog, the surface area will be closer to 3, but the volume is still 1. You could continue this until your object is more like a very long strand of spaghetti. The surface area might be 40, but the volume is still 1.
Image file compression lowers the volume (number of bytes) while leaving the area (height and width in pixels) unchanged. It keeps the number of pixels the same, but lowers the number of bytes used to store the file.
You specify the image height and width in the "Size to" drop-down of the Image Export dialog.
You don't specify the storage requirements (which is why you have to find out what they are after you export). Instead, you specify two things: the file format and the "Image Quality". These two, combined with the number of pixels (the height x the width) will determine how much storage space your file requires -- how many bytes "big" it is.
File size is measured in bytes.
Image size is measured in pixels.
So your original question is more accurately stated:
I have an image in my Aperture library that has a file size of 18.1 MB. I exported out using the original Image size - height and width in pixels - and saved it to my desktop. When I checked the size of the file I created by exporting, it bumped it way down to 1.8mb. How did that happen? I then opened the same image file inside of Photoshop for some editing and exported it back out, overwriting the file. When I checked the file size again, it had dropped to 1.2 mb! What is causing my image to lose the number of bytes used to measure its size?
And the answer is: the file format you use and the amount of compression.