The preview application has a mode in its Instant Alpha selection tool that makes it easy to select the background. After selecting the background image, you can simply press the “Delete” key and this area of the image will be deleted. Save it as a PNG with the "Alpha" flag and you're done.
Most other image editing tools, especially those that support multiple layers in the image, will also correctly support PNG alpha channels. Most likely, it will be easier than using Preview, but they are all third-party and many of them cost money, so you will need to make this decision.
Please note that if you start with an image with a solid background, to which some things have a faded edge that merges with the background, it is unlikely that you can completely remove the background to your satisfaction without cropping the actual image.
You may have seen GIFs in the past that were designed to blend on a white background and had bright pixels at the edges that stood out and looked ugly on darker backgrounds. An image saved as PNG or any other format with an alpha channel from the very beginning will not have this problem.
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