Browsers typically cannot access the OS clipboard - it would be a disaster if a web page could get access to your credit card number, that you happened to have copied, for example.
Therefore, web apps cannot copy/paste to the OS clipboard, unless the user actively presses Ctrl-C/Ctrl-V or similar.
This makes copying/pasting spreadsheet data difficult, since it is not just a block of text or an image, but a more complex data structure (cells with content and formatting). To work around this, Google has implemented a replacement clipboard, which stores clips on their servers instead of in the OS clipboard. This is the web clipboard (also called Server clipboard or Cloud clipboard).
To paste from the web clipboard, click on the clipboard icon in the toolbar - don't use the Edit menu - and your copied item should be listed there. See this blogpost for explanation.