Some websites show the company name next to the URL (in Chrome) and some don't.
How do I set this up for my website? Is there a reason why or why I wouldn't want to do this?
This is only for websites that use Extended Validation SSL encryption. The name is the name the certificate is issued to. It is a security feature put in place to verify that the website is actually served by the company it claims to be from. Simply put, it works like this:
You need a couple of things to set this up, most notably:
Certificates generally don't come cheap, and it is really a matter of whether you really need it. Admittedly, it looks cool, but it might be unnecessary for you.
You can find more detailed information about this from other sources, like its Wikipedia page.
That's a result of using an Extended Validation certificate - extended validation, or EV certificates require additional validation before the certificate is issued. The idea is that there is better proof that the company really is applying for the certificate, so you should be able to have some more confidence that you really are talking to who you think you are.