This page deals specifically with the Facebook Profile Image. I also have a more detailed guide to using images on Facebook.
If the Cover Photo is the place to make a good first impression on your Facebook page, the Profile Picture is what represents you. It displays prominently on your page, but it also follows you everywhere you tread on the site.
Most people use a photo of themselves--and that's really what it's designed for--but you can use any image you like, even a logo. Some people use a photo of their kids. Some use it to make a political statement.
It used to be that the cover photo was overlaid with the profile picture in the bottom left, but in the latest design they've been separated into their own spots. Like this:
In choosing a profile picture, choose something that will work well at small sizes. While the one that's displayed with the cover photo is a reasonable size, the same profile picture is also used much smaller elsewhere on Facebook like next to any comments you post. Also, remember that your profile picture is always public (as is your cover photo).
It's always displayed as a square. You can upload a rectangular image, but it will be cropped to a square shape.
It will be displayed at 170px by 170px on computers and 128px by 128px on smartphones.
The thin white border is added automatically, and there's no way to remove it.
Profile Picture on the Timeline
The profile image that appears next to your name on comments and posts is the same image but is automatically scaled down to 40px by 40px.
Uploading a New Profile Picture
If you use a photo that's not square, you have some control over which part of the image to use for the crop. When you're logged in, hover over the profile image and choose "Edit Profile Picture" and then "Edit Thumbnail."
You'll then get a basic cropping tool. Drag the corners of the box to change its size. Click inside the box and drag to choose a different part of the image.
You can then click on the thumbnail and drag. Click "Done editing" when you're happy with it to go back to your page.
Tip: If you find that your resulting profile picture is blurry after it's downsized in Facebook, try uploading an image twice the size of the downsized image (that is, an image that is 340px by 340px). That often gives a sharper result.
The Profile Picture is automatically overlaid on the Cover Photo. It's 22px from the left edge of the Cover Photo and 211px from the top.