Separating images is one of the basic skills. In case you are new to Photoshop, this is a good way to familiarize yourself with your selection tools and layers. Learning how to separate pictures will teach you how to use shortcuts and make accurate selections. It is possible to edit, cut or separate anything inside the moving dotted line. In case you feel comfortable selecting the image you want to separate, go to the section to separate the image. There are a variety of different selection tools available, each with its pros and cons
In case you want to make a new composition using a layer, you just need to click and drag. In case the layer is already separated, simply click and drag it out of the background. You can take it to Illustrator or create your own Photoshop layer. You can also delete all other layers and use the "Save As" option.
In case you want to extract an image from your background, but do not leave a huge gap where the image was, you'll have to find a way to replace that space. It may be easier said than done, depending on the background. In case you have a basic and basic background like grass or ocean, the "Clone Stamp" tool will be able to copy some section of the image and use it to paint over the hole.