As a general rule, you send an entire Chrome browser tab to your Chromecast (or other Cast-ready devices) as a last resort. The quality is lousy enough that you only really want to use it when a service doesn't have native casting support, like Amazon Prime Video. Google may soon give you a good reason to use this option more often, though. The latest developer version of Chrome includes rough support for a feature that preserves the quality and performance of video when you cast a tab. If you choose to take a video full screen while casting, the software now sends the video's raw bitstream to your TV. You shouldn't get muddy, stuttering video as your PC struggles to deliver content you don't need to see.
In this crude implementation, you'll have to enable the video improvement through a flag (chrome://flags/#media-remoting). Google's François Beaufort suggests using Vimeo to give it a try, but Android Police has managed to get it working with Amazon, the Cartoon Network's Adult Swim website and other services. There's no guarantee that this will reach more polished versions of Chrome in a timely fashion, or at all. However, it's easy to see Google pushing for this feature as a way of filling a conspicuous gap in Cast support -- imagine seeing almost any video on your TV without having to wait for the video's provider to get with the program.