There are a myriad of ways to separate an image from a background. In this example, we are going to be using the pen tool and create points along the parts of the image we wish to extract and place onto another background.

Use the pen tool (you could also use the freeform pen tool, if you feel more comfortable) to draw an outline around the image. You don't have to be 100% accurate in making your point selections, however, it is better to err as far as cutting into the image versus leaving a gap. Continue along the edges until making the final point to close the path.

When you are finished, right click and click on Make Selection.

Enter a value of "2" for the feather radius.

Control + Copy to a new image.

It is just as easy to incorporate your selection into a background of a different color or theme. It is not always necessary, although the opacity of the copied layer has been lowered slightly here to offset any possible blending problems. Notice no jagged edges.