The simplest way to do it, is to use javascript (client side).
Below is a very crude example on how to do this. This will give you an idea on how to proceed.
create a html page, with two separate text area boxes.
Let the first text area box be the source where you type in the text.
Assign it an id 'source_area'.
<textarea id='source_area'>
</textarea>
Let the second text area box be the destination.
Assign it an id 'destination_area'.
Set this area as "readonly" because you don't want users typing here directly.
<textarea id='destination_area' readonly>
</textarea>
Now when a user types into the first box, we need to capture the particular action.
For this example I will use the "onKeyUp" to capture events when a keyboard key is released.
Now when typing into the source text box, a key on your keyboard is released, it will invoke a javascript function "transferToNextArea()" is invoked.
We will create the javascript function "transferToNextArea()" in
Read more about javascripts here.
http://w3schools.com/js/js_events.asp
Complete list of events here.
http://w3schools.com/jsref/dom_obj_event.asp
The javascript function will extract text from 'source_area' text box.
It will then assign the same text into 'destination_area'.
function transferToNextArea()
{
//extracting text.
var varSrcText = document.getElementById("source_area").value;
//assigning text to destination.
document.getElementById("destination_area").value=varSrcText
}
Complete html (tested in Google Chrome)
<html>
<body >
Source Box
<textarea id='source_area' onKeyUp="transferToNextArea();">
</textarea>
<br>
Destination Box
<textarea id='destination_area' readonly>
</textarea>
</body>
<script type="text/javascript">
function transferToNextArea()
{
var varSrcText = document.getElementById("source_area").value;
document.getElementById("destination_area").value=varSrcText
}
</script>
</html>
This is just a very basic example. It is not very effecient, but it will give you an idea of how data can be moved around.
Before assigning the text, you could manipulate the text however you want it using javascript.
Stackoverflow formats the text as per the html tags after extracting it. This will require lot more code and more work.
Using a servlet for the above task is overkill.
You would use a servlet, only if you want to do something with the data on the server side.
Example
a) store it in a database before displaying it below.
Read about "ajax" calls to send and recieve data between the server and client.
Ajax will give you the means to send data to the servlet without having to refresh the whole page.