A transparent background means part of an image has no colour at all — instead of white or a solid box, you see whatever is behind it. This is essential for logos, product photos and stickers that need to sit cleanly on any page or design.
Only certain formats support transparency: PNG and WebP do, while JPG does not (which is why JPG logos end up with a white box).
You can create transparency by removing the background from a photo, then save it as a PNG. To make a round avatar, use Circle Crop.
How transparency is stored
Formats that support transparency add a fourth channel, called alpha, to the usual red, green and blue. Alpha records how opaque each pixel is, from fully transparent to fully solid. Because it is a range rather than a switch, edges can be partly transparent, which is what makes anti-aliased text and soft shadows sit cleanly on any background instead of showing a jagged fringe.
Which formats support it
PNG and WebP support full alpha transparency and are the formats to use. GIF supports only one fully transparent colour, with no partial values, which is why GIF cut-outs show hard, stair-stepped edges. JPG has no transparency at all — converting a transparent image to JPG fills the empty area with a solid colour, normally white, and that fill cannot be undone afterwards.
The checkerboard is not part of your image
Editors and preview tools draw a grey and white checkerboard to indicate transparent areas, because there is no other way to show "nothing" on screen. It is never saved into the file. If a checkerboard pattern appears in your exported image, the image itself was flattened onto a checkerboard graphic at some point — usually by a screenshot rather than an export.
Why a cut-out sometimes has a white halo
When a subject is cut from a light background, some edge pixels blend the subject with what was behind it. Placed on a dark background afterwards, those partly-light pixels read as a pale outline. Contracting the mask slightly, or re-cutting from a source where the subject contrasts with its background, removes it. This is also why background removal works best on images where the subject is clearly distinguishable from what sits behind it.