Client-side rendering, on the other, is rendered with JavaScript. Instead of getting the HTML by itself, you're getting a simple HTML structure but with JavaScript to render the rest of the HTML with your browser.
This is a new approach and has gained a lot of traction lately with the development of framework from companies such as Facebook and Google.
The main difference is that when you click to take a look at another page, the JavaScript renders the page instead of requesting a new page from the server. This is a much faster way to load content, as it will only load content that it needs instead of the whole page.
The downside of it is that it's very bad SEO-wise, since the content is not rendered until the page is loaded into the browser. This can be fixed, but not in an easy way yet.