To use Chromecast on your mobile device, make sure you're connected to WiFi, and open a supported application, such as Netflix, YouTube, Hulu Plus or Pandora. Installation of the Chromecast app (which you already did for setup, most likely) makes a special Cast icon appear within supported apps. You may see the icon as soon as you enter the app, or it may appear once you select a video, song or other media to cast. When you click the icon, it prompts you to select a Chromecast (in case you have multiples) and, after a moment, your chosen media should appear on the TV.
In the Chrome browser on a computer, once you've installed the Cast extension, a cast icon will appear in the upper right-hand corner or your browser that lets you cast a single tab to the screen. You also use the icon to stop casting the tab. There are some Web sites, like Netflix and YouTube, which are already optimized for Chromecast. They will include a Cast icon on each video that allows you to rather seamlessly cast the media to your TV rather than the tab itself, which should result in higher quality and fewer playback glitches. Video cast from a Chrome tab has a maximum resolution of 720p, whereas video streamed via the Cast button for a video housed on a supported site can reach 1080p.
You generally control streaming from your mobile device using a volume bar, a scrub bar (the scroll bar that lets you move forward and backward through a song or video), and stop, pause and play icons on the screen. There may be other controls available, depending upon the app. The volume in this case is Chromecast's internal volume, so you may have to use the TV's remote to really get the volume to your preferred setting. If you have an Android device, you can also control your streaming from the lock screen or notification bar.
You can leave the control screen and browse within the app, go to other apps, let your computer or mobile device go to sleep and even shut it off, and your chosen entertainment should keep playing on the TV via the Chromecast. The latter cannot be done when casting a Chrome tab, but it does work with Netflix and the like.
While you're streaming a video via Chromecast, if you select another video from within the same app, that video will likely play to your TV screen instead. But you can actually go to another application and play a different video on your mobile device or computer while the original video continues to stream on your TV screen, unless you choose a cast icon in that app (if available), in which case the video in the new app will take over your TV. If you have wandered off and want to take control of a streaming video to once again, you go back to the app, and there should be a "Now Playing" bar or similar that takes you back to the control screen for that video.
Multiple devices can be used to launch content to the Chromecast, provided they can all connect to your WiFi network and have the appropriate apps or extensions installed. They can include your phone, your tablet, your computer or the devices of other household members or friends who are over. That means you can end up fighting for control of the TV, or you can purposefully do a bit of organized social streaming of videos or music.