After you remove unwanted regions of the image, save in a format that supports transparency. Photoshop PSD or PSB, TIFF, PNG or GIF, for example.
Are you trying to save a portion of an image with a transparent background? It may help to post a screenshot of the image that you're working with as well as your layers panel.
I've included a few tips below that may help you with your situation. If you need additional help, please feel free to post again
Deleting a background and saving an image with a transparent background
1. Here I am working with an image of a santa hat that I've cut out from a white background. I have two layers, the cut-out santa hat and the original file. I'm going to delete the background layer so that I have just one layer - the cut-out santa hat:
2. You'll see that a checkerboard pattern has appeared behind the santa hat. This means that the background is transparent.
3. You can then go to File > Save As... and select PNG, Photoshop PSD, or TIFF from the Format dropdown menu. These file formats will save the cut-out portion of your image on its own, while preserving the transparent background.