Reading the <textarea>‘s content:
var text1 = document.getElementById('myTextArea').value; // plain JavaScript
var text2 = $("#myTextArea").val(); // jQuery
Writing to the <textarea>:
document.getElementById('myTextArea').value="new value"; // plain JavaScript
$("#myTextArea").val('new value'); // jQuery
See demo JSFiddle here.
Do not use .html() or .innerHTML!
jQuery’s .html() and JavaScript’s .innerHTML should not be used, as they do not pick up changes to the textarea’s text.
When the user types in the textarea, the .html() won’t return the typed value, but the original one — check the demo fiddle above for an example.