Breaking down the buzzword: what is cloud computing in regards to Chrome OS?
Cloud computing is commonly used as a generic term for the shift from local computing and data storage to remote solutions. To simplify, it means a move from keeping your files locally to putting them online. There are real benefits, but as with any opportunity, they aren't entirely free of potential drawbacks.
Chrome OS is considered a web-centric operating system, one that works best when you have a strong Internet connection. That isn't meant to imply that it doesn't work offline, but it will work better when you do have access to the web. This is partly because of Chrome's heavy reliance on Google's Cloud technologies.
Chrome has a powerful sync functionality under the hood. It works by making a copy of your local profile settings (profile photo, bookmarks, wallpaper selection, etc.) and putting it securely on Google's servers. Your device periodically connects to those servers and updates the remote copy to keep it "in sync". The benefit of this is that your settings are backed up, and always up to date. If your Chrome device is stolen you can purchase another, sign in, and all your settings will instantly be restore. These settings can be managed on the chrome://settings/syncSetup page.
The File Manager included with Chrome OS has a folder dedicated to Google Drive, Google's file backup service. If you drag and drop a file from the "Downloads" folder into the Google Drive folder, it will instantly sync in the background. This allows you to easily keep your files backed up remotely, so if your device breaks you won't lose them. Google Drive can be accessed from almost any device with an Internet connection, including most smartphones.
This isn't a yes or no question, and any source that claims it to be that simple isn't being entirely honest with you. As a general rule, no system should ever be assumed to be entirely secure. Hesitation should be used whenever you are working with sensitive data.
However, Google makes every effort to securely store your data once it has been saved to their servers. The same can be said of most other legitimate technology companies, they all have an interest in keeping user data as secure as possible.