Every tweet sent has its own web page with an address (URL) that will point to it. You can share this link in a tweet of your own, how the tweet appears will depend on what app or platform other readers are using.
On Echofon for iPhone the tweet will appear as your text plus a clickable link to the tweet but on desktop Twitter the system will convert the tweet’s link into a visible embedded tweet that shows up without having to click on the link or tweet to view it. I believe this is the same on Twitter for iPhone etc. It will also appear in the Notifications tab of the original tweet sender.
It’s important to be aware of this as people have previously used this technique to ‘sub-tweet’ (ie send a critical tweet about someone else’s tweet without that person being aware). Fortunately, or unfortunately, Twitter changed how such tweets were handled and this is now more easily discoverable. Previously they’d have had to search for tweets mentioning their username, eg by searching username or twitter.com/username).
Avoid this by… using http://www.donotlink.com/ to hide the tweet’s link, then it will not notify the tweet’s author.