What is the easiest way to remove background from below image?
If you can show it with screenshots it would be great.
You could use the Quick Selection Tool (W) or the Magic Wand (W). Quick Selection will probably be better as you control, with your mouse, where and what you want to select. The magic wand is a one click solution that is better for solid areas of colour.
Once you have selected all around the jewel you can invert your selection Select/Inverse and delete the background then refine your edge Select/Refine Mask.
Sorry don't have time for screenshots but here is a tutorial- http://www.photoshopessentials.com/basics/selections/quick-selection-tool/
deecemobile's suggestion works, but if you want a super clean edge, you may have to trace it with the pen tool and create a vector mask.
As a third option that I've had mixed success with there is a "Background Eraser Tool" that can in theory remove the background quickly.
Though for the best success I'd use vector or raster masking.