With a few exceptions, most of the key adjustments you can apply to an image can be done in two ways: from the Image menu under Adjustments, or as an adjustment layer. Very simply, an adjustment layer gives you the ability to return to the original adjustment dialog to make further changes, whereas the Image>Adjustments commands do not. Here's a comparison of using Levels in two different ways.
First, I pressed Command-L (PC: Control-L) and made an adjustment to the photo. Looking at the Layers palette, it's clear that the adjustment was made directly to the Background layer. If I were to save that document and then open it weeks later attempting to tweak the Levels command, I'd get a Levels histogram that looked something like the one you see here. This kind of histogram indicates that changes have already been made to the image, and although it's theoretically possible to make further changes, it is challengingeven if all you want to do is return to the original image.
In contrast, choosing Levels from the Create New Adjustment Layer icon's pop-up menu (at the bottom of the Layers palette) adds a new layer to the document. At any time, I can double-click on the adjustment layer's thumbnail to re-open the Levels dialog and get the original dialog, just as I left it. This allows for ongoing adjustmentsas long as you save a layered document that includes the adjustment layer(s).
The other advantages of an adjustment layer include the ability to hide it, delete it completely, change its opacity or blend mode, or include more than one of the same type of adjustment. So, am I suggesting that you should always use an adjustment layer? Not necessarily, but I would suggest (insist?) that you at least consider using an adjustment layer to determine if that would be the best way to go.
One last thing: there are a few types of adjustments that are not available as an adjustment layer (such as Shadow/Highlight, Match Color, and Exposure). In those cases, the next best thing to using an adjustment layer is to duplicate the Background layer and apply these commands to the copy of the Background. That way you can at least experiment with opacity and blend modesor throw away the layer and start over again.