The first thing you need to do is to get the image you want for your favicon (surely not stating the obvious... eh?;) ). It's better if it's a more "squared" size, because you just can't have a rectangular favicon. If it's on Internet, just save it on your computer.

Alright, so after you've saved it, head over http://www.favicon.co.uk/ and browse for your image on your computer. Then choose any size you want, it doesn't really matter, however I prefer 32x32 (dunno why..). Now click on Generate Favicon. Wait a little bit, and you should get this:

Click on Download Your Favicon Here and it'll be saved on your computer.

Next, you'll need to upload that icon to your website.

P.S.: You can also try favicon.cc where you can draw your own favicon or create animations! (Credits to Pobman for this add)