It explains in the Droid manual you linked above on page 24 how to do so, and even then it's easy enough to figure out. I've only had my Droid for 24 hours and never touched an Android device before.
Firstly, cut/copy/paste/delete only work in text input fields that I've noticed. That means in email, SMS/MMS, contacts, search boxes, web browser address line, etc. I haven't gotten it to work in a way where I can copy/paste text from a web site or anything like that.
This is the step by step process:
1) For all of the text in an input field, skip to step 3.
2) For only specific text in an input field, you HAVE to be in landscape mode with the physical keyboard open. Then, just like on a computer, you click on a spot in the text input field that is to the left of the text you want to copy, hold SHIFT (up arrow), and then keep tapping the D-pad's right side. If you want, you can click a spot to the right of the text you want to cut/copy and tap the D-pad to the left as well. Maybe if it's a paragraph of information you will want to move up or down on the D-pad. Whatever it is, select the text by holding shift and tapping the D-pad as desired.
The only difference from a computer is that you use a D-pad here instead of the arrow keys, but the D-pad in essence is made up of arrow keys with an Enter button in the middle.
3) Tap and hold the input field/selected text until a context menu pops up.
4) Select "Cut" or "Copy" as desired.
5) Tap and hold your finger on the destination input field until a context menu pops up.
6) Select "Paste."