I notice a lot of cool sites being listed there with great stats and archived layouts and details and whatever, even screenshots.

How do I get my site on alexa and how do I get my details on it too, I couldn't really find anything I really understand

If you're listed in google, you'll be listed in alexa. They do have additional (to google) results sometimes, I'm not sure where they get them from, probably data obtained from alexa toolbar users.

Once you're site is listed there, you can visit the page for your site and there's a link to update the screenshot/thumbshot.

I wouldn't worry about alexa too much, the stats are obtained from alexa toolbar users only so they are far from accurate.

You can enter your own contact information, it will then ask you to verify it after which it will take a few days before it goes public. I think it will get the screenshots while browsing it frequently.

you can click update thumbnail and it will grab a screenshot within 48 hours