All social media currently show profile pictures within circles, but no camera that I know of yet takes images in this format! You can generally upload rectangular images and then adjust the central point online, however it may help to first use photo editing software to crop your image to a square format, making sure that it looks just right before uploading.
For a headshot or profile picture, remember that "the eyes have it"! So when choosing your image, make sure that these are in sharp focus. Ideally there should also be a bright reflection in each eye as this gives a lively "sparkle" to your expression! Photographers refer to these reflections as “catchlights” and will often use extra lights or a reflective board to provide stronger reflections.
For a strong image, you may want to crop in tightly to the face. This reduces any distractions in the background, whilst ensuring attention on the real subject of the picture – ie, you! Remember that profile pictures are relatively small images on screen, so you don't want to appear as an indistinct shape!
Cropping in tightly also enables you to take advantage of the "rule of thirds". This is a compositional guideline, well known to artists since the 18th century, which suggests that an image can be more aesthetically pleasing if key elements are aligned on the 1/3 lines. The idea has since been applied to everything from paintings of landscapes to PowerPoint presentations!
If you are uploading a profile picture to LinkedIn, you'll see four guide lines that make it easier for you to apply to rule of thirds to your image. On other social media, you can still bear it in mind – again, cropping to a square format before you upload will be helpful. Try adjusting your image so that the eyes are aligned with the top horizontal guideline. Even better if you adjust the size of your crop so that the two eyes also align with the two vertical lines - these intersections are often called "power points", for good reason. Zooming in this tight is likely to mean cropping out the top of the head, which may seem scary... but the result can be an image with a modern look that emphasises your eyes and ensures a strong connection to the viewer.