That image is called 'favicon' and it's a small square shaped .ico file, which is the standard file type for favicons. You could use .png or .gif too, but you should follow the standard for better compatibility.

To set one for your website you should:

  1. Make a square image of your logo (preferably 32x32 or 16x16 pixels, as far as I know there's no max size*), and transform it into an .ico file. You can do this on Gimp, Photoshop (with help of a plugin) or a website like Favicon.cc or RealFaviconGenerator.

  2. Then, you have two ways of setting it up:

A) Placing it on the root folder/directory of your website (next to index.html) with the name favicon.ico.

or

B) Link to it between the <head></head> tags of every .html file on your site, like this:

<head>
  <link rel="shortcut icon" type="image/x-icon" href="favicon.ico" />
</head>

If you want to see the favicon from any website, just write www.url.com/favicon.ico and you'll (probably) see it. Stackoverflow's favicon is 16x16 pixels and Wikipedia is 32x32.

*: There's even a browser problem with no filesize limit. You could easily crash a browser with an exceedingly large favicon, more info here