Cloud Clipboard is one of the features which wasn't included with Windows 10 Fall Creators Update. It is a special feature to sync clipboard content across all your devices connected with your Microsoft Account. It is powered by their Cloud infrastructure and utilizes the same technologies that made it possible to have your preferences synchronized across your devices, and your files made available everywhere with OneDrive. The Cloud Clipboard made its appearance in the recently released build 17004.
The well-known Windows enthusiast, Rafael Rivera, managed to get it working.
According to his research, Cloud Clipboard has a global hotkey Win + V assigned to open a special pane. It lists the items in the user's cloud clipboard along with local items. For items stored in the cloud, it shows a tiny cloud icon. Here are some screenshots.
As you can see above, the Touch keyboard app also seems to have received support for the cloud clipboard feature, so tablet users can quickly access it with one tap.
The Cloud Clipboard feature doesn't work properly as of this writing. A working version of this feature is expected in "Redstone 4" builds of Windows 10, which will be available to Insiders in a couple of months.