The featured image is displayed in connection to the post or page, usually on both archive pages and single pages. Some themes display the caption of the image as well. When your post or page is shared in social media, it’s usually the featured image that is displayed in the preview for the web page. Support for the featured image functionality is included in most themes, but not all of them.
How big the featured image you upload should be depends on which theme you have, but since most themes automatically crop and resize an image to the appropriate sizes, it’s better to upload an image that is too big rather than one that is too small.