How to lighten photo on right side only with a fade?

Apply a mask to the half you want to lighten, then feather the mask - the portion that hangs over the edge of the photo will feather too, but will not matter as it is past the edge (just make sure it is larger by a sufficient enough margin).

Now you have two choices. Many times the effects you apply will only apply to the masked portions of your image, but sometimes they effect the entire document. A more sure fire method that will probably yeild more controlable results would be to make a new object from the masked portion, then do all of your lightening and brightening to the duplicate. It would be easiest to hide the original image, then make your adjustments to the duplicate. Since you won't move it from its location, when you make the original image visible again, the effect will appear to blend on the same image.

In retrospect, an easier method might be instead of feathering the mask, just make the mask with a hard edge, duplicate the masked portion, then use the Interactive Object Transparency Tool tool to manually create you fade. This will allow the added benefit of being able to fade in multiple locations/directions if needed.

The mask/feather method attrofy posted is the classic way to do that effect (mask the half you want to light, feather the mask) but for all effects...
I mask/select the portion I want to "effect"
I copy/paste as new object or selection with a transparent background
Apply effects to that object (portion)
Then merge all objects after I done the effects..
Keep a clean copy of the "object" somewhere...either in the stack as "invisible" or as a separate "copy from clipboard" and apply affects to duplicate copies
That way if mask/feather doesn't work, perhaps a transparency, layer blend mode or a filter would be a better approach..You have endless copies to play with..if you save a clean copy of that portion..