User testing is taking people to your website to see how usable a website is. They will usually be given a list of tasks to complete, and they note how the website performs in completing those tasks.
It also allows the website creator to be able to gain feedback on changes they might not have even been aware of, and implement them into the website. It can be a common occurence when the website developer doesn't notice any issues that a new user would, as the developer has looked over the website and code tens or even hundreds of times.
Cross browser testing is a test to make sure that your website can work on different web-operating systems, like Firefox, Chrome, and Safari. It also allows the website to be testing for different devices, such as a phone, tablet, tv, and a computer.
The testing is done on two factors, popularity and analysis. Popularity refers to picking a dozen of the most commonly used browsers and test your website to see any inconsistencies or bugs. Analysis refers to using browser tools to see what kinds of browsers you users and using to access your website.
www.browserstack.com
www.hotjar.com/usability-testing/