I think you need to clarify the question a little, or give an example of what you're looking to do. I'm a web designer/developer and could likely answer the question. Sorry.
How do people put there leopard geckos on there webpage and it is just there leopard gecko? Not the background of the image they took, just the Leo. For example on Enigmatic_Geckos avatar there is just a plain leopard gecko with no image of the background.
There are many different ways to do it. Email me and I can help a little. There are so many ways of doing things, after you have had experience with website and graphic design you learn the best way to do things based on what exactly you want.
you can use a tool like photoshop or gimp to cut out the image, then take it past it into a new image with a transparent background. edit, resize if you want and save as a gif, or png
crop out the gecko from the original picture and paste it to the background image that'll be used on the website using photoshop or another graphics program.
Just put picture in photoshop! with the pen tool select what you want in a image! and delete the unselected part (background)! transform to PNG format with a transparent background setting!