Up to Photoshop 7 there are two types of layers -- image layers (with actual stuff on them) and adjustment layers (which contain settings that affect the appearance of the layers underneath them) [for the purposes of this discussion and so as not to get too confusing too quickly I am classing things like text layers and shape layers as "image layers"].
To envisage how this works, it's easiest if we start with the image layers.
Have you ever seen a documentary or read an article about old-fashioned cel animation? In that you get the background and then the characters are drawn on clear acetate and layered over the background. The characters can then be moved around or changed without having to redraw the background every time. Image layers are like that -- think of them as being bits of image on clear acetate.
Adjustment layers are a bit harder to get your head round. You can make an adjustment like changing the Levels to your whole image (or one of its image layers) but that is a permanent change and if you decide later that you overdid it there's not much you can do about it. Instead, you can use a Levels Adjustment Layer -- think of this like a magic bit of acetate that changes the appearance of the layers below it you had changed their Levels, but because it's an adjustment layer you can always get rid of it or turn it off later.
Other things you can do with layers:
- Make an individual layer visible or invisible
- Adjust the opacity of an individual layer
- Use Layer Masks -- if you add a mask to a layer you can selectively make parts of that layer invisible without having to permanently delete anything
- Use Layer Blending Modes or Layer Styles -- I won't get into those here as that's probably something to get into once you're more confident with layers in general.
One introductory tutorial on this site you might want to try is here