If you are trying to copy a URL, first load that URL in your browser, then long-press the address bar and select 'Copy page url'.
If you want to copy text from a web page, then hit Menu > More > Select text (or Menu+E on the hard keyboard), then drag your finger to highlight the text. When you pick up your finger, the text will be copied to the clipboard automatically.
To cut, copy, or paste stuff into a text entry box, just long-press the box and select the item corresponding to what you want to do.
There are also keyboard shortcuts for a lot of copy/paste tasks. Eg, you can use shift-Dpad to select text, Menu-X to cut, Menu-V to paste, etc.