For example you find a gif from the internet that you like but there is a background stopping you. Well you can remove it using Photoshop.
Heres how.
Start with your gif. I will be using this one:

Open Photoshop and go File>Import>Video Frames To Layers and select the gif.
You will get a menu like this:

Type *.gif as the name and you will be able to see all your gif files.
Select your file and continue to hit ok.
Now if you can’t already see the Animation window go Window>Animation. You will use this later after editing it.
I will simply use the magic wand tool to select the background and delete it on the specific frames.
Now you have finished press the drop down menu of the animation window and press make frames from layers.

Now go File>Save For Web & Devices… and if your happy with it press save and put it somewhere.
Congratulations!
My end product:

This method can be used to incorporate gifs into other images.
Enjoy!
My end product: