Let me start with this picture:
In this case I want more space at the left side of the picture. I can do that using the crop tool (C):
But make sure that "delete cropped pixels" is off. That results in this image:
Next step is to fill the transparent part of the picture. I can do this by first selecting this part of the image with the Magic want tool (M). When selected you've got to fill it. Go to Edit > Fill and choose Content aware.
Hit Ok and see to it that Photoshop is able to fill it correctly. In this case it's not perfect but not to worry, it's easy to correct (with the Clone Stamp (S) and Spot healing brush tool(J)).
With a bit of extra work, this is my end result:
The team of Photoshop is now working on a new feature called Content Aware Crop. Check out this sneak peek to get an idea of how it's supposed to work: