I always wondered, also, Youtube.co has the youtube logo then Co next to the logo on the site.
The co a country code. I’m not sure which country. Columbia?
Yes. It is registered to Google, so it’s probably a typosquatting prevention registration. And also convenient for Columbians. 
There’s also Youtu.be , which is technically a Belgian address.
The part of a Web address after that last dot is the top-level domain. Every country has their own two-letter top-level domain (most of the non-country ones are three letters), and can decide for themselves what criteria they want to use for registering websites in that domain. In practice, most countries have decided that the criterion they want to use is just if someone pays them a nominal fee for it. And a lot of Web sites have decided to register with those countries’ top-level domains just because they like the pair of letters it happens to give them. Probably YouTube registered the .co version just to catch anyone who mis-types .com . And the .be one is because it makes the URL shorter, which is convenient when you’re texting or tweeting a link.
Oh, and the country in question is Colombia, not Columbia.
Where I am, google.com redirects to google.ca. I think google.ca behaves more Canadianly but I’m not sure what I mean by that. Sometimes the doodle is different.
I’m an engineer. I’m not expected to be able to spell or use proper grammar. 
“I’m feeling lucky, eh?”
Maybe it’s more polite?
Before Youtube became so good at identifying you and remembering what you last viewed (even if you don’t log in), it used to show you different content depending on which portal you logged into.
So if you logged into the UK site, you’d see UK material. the AU site, Australian material. And sometimes people from Columbia wanted to see Columbian content, and liked to see the logo on the site identifying that they were on the Columbian site, not on the foreign site of some big foreign company.
I have to say, this is much less common now than it was 15 years ago. But it used to be that if you had different national sites, it was something that you bragged about.