JPEG uses lossy compression, which is visible pretty much always if you know where to look for it. You can always tell an image has been stored as a JPEG or similar format.
JPEG does not store pixels directly (the way the TIF format does), but rather does some form of Fourier transform on the image and then merely stores the different "frequencies" present. In general you will need to store frequencies with wavelengths twice as high as the distance between the light-sensitive elements of your CCD in order to encode ALL detail you camera captures (Sampling Theorem). Pretty much noone does that, as that is not much different than storing the file in TIF or RAW, but you can throw out some high frequencies without degrading the image in a very perceptible way. In Fine mode you will throw out fewer of them, and thus encode more of the detail within the image, while in Normal you might throw out more and end up with less-detailed image.
Here is an example:
Suppose you are taking a picture of a basketball. Normal mode might not store that the ball has black stripes, as this is hardly visible from the distance you are shooting at, and thus will make your basketball look more like an orange sphere, while Fine will keep more of that detail and you will have an easier time saying "That's a basketball". Looking at the two pictures, however, they will both bear the general shape of a basketball, but one will have slightly more contrast and detail.
The devil is in the details, as they say.
Even if your picture looks acceptable straight out of the camera, you will still see "ringing" effect of areas of high contrast, like edges of things, as well as in places where there are very subtle differences in color, like areas with shadows. These are called compression artifacts, as previously mentioned, and they become more and more pronounced the higher the compression (the less detail you store). To make matters worse, if you post-process your images, you will end up uncompressing your image, altering it in some way, and then recompressing it, meaning that the compression artifacts of the first save start getting compression artifacts of their own. Thus every time you resave a JPEG, you degrade its quality.
I hope this helps you.