Click button copy to clipboard

This is how it works:

  1. Creates a temporarily hidden text field.
  2. Copies the content of the element to that text field.
  3. Selects the content of the text field.
  4. Executes the command copy like: document.execCommand("copy").
  5. Removes the temporary text field.

NOTE that the inner text of the element can contain whitespace. So if you want to use if for example for passwords you may trim the text by using $(element).text().trim() in the code above.
You can see a quick demo here:

function copyToClipboard(element) {

  var $temp = $("<input>");
  $("body").append($temp);
  $temp.val($(element).text()).select();
  document.execCommand("copy");
  $temp.remove();
}
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.11.1/jquery.min.js"></script>
<p id="p1">P1: I am paragraph 1</p>
<p id="p2">P2: I am a second paragraph</p>
<button onclick="copyToClipboard('#p1')">Copy P1</button>
<button onclick="copyToClipboard('#p2')">Copy P2</button>
<br/><br/><input type="text" placeholder="Paste here for test" />

Update 1: This can be achieved also with a pure JavaScript solution (no jQuery):

function copyToClipboard(elementId) {

  // Create a "hidden" input
  var aux = document.createElement("input");

  // Assign it the value of the specified element
  aux.setAttribute("value", document.getElementById(elementId).innerHTML);

  // Append it to the body
  document.body.appendChild(aux);

  // Highlight its content
  aux.select();

  // Copy the highlighted text
  document.execCommand("copy");

  // Remove it from the body
  document.body.removeChild(aux);

}
<p id="p1">P1: I am paragraph 1</p>
<p id="p2">P2: I am a second paragraph</p>
<button onclick="copyToClipboard('p1')">Copy P1</button>
<button onclick="copyToClipboard('p2')">Copy P2</button>
<br/><br/><input type="text" placeholder="Paste here for test" />

UPDATE: COPY KEEPING THE TEXT FORMAT

As a user pointed out in the Spanish version of StackOverflow, the solutions listed above work perfectly if you want to copy the content of an element literally, but they don’t work that great if you want to paste the copied text with format (as it is copied into an input type="text", the format is “lost”).
A solution for that would be to copy into a content editable div and then copy it using the execCommand in a similar way. Here there is an example – click on the copy button and then paste into the content editable box below:

function copy(element_id){
  var aux = document.createElement("div");
  aux.setAttribute("contentEditable", true);
  aux.innerHTML = document.getElementById(element_id).innerHTML;
  aux.setAttribute("onfocus", "document.execCommand('selectAll',false,null)"); 
  document.body.appendChild(aux);
  aux.focus();
  document.execCommand("copy");
  document.body.removeChild(aux);
}
#target {
  width:400px;
  height:100px;
  border:1px solid #ccc;
}
<p id="demo"><b>Bold text</b> and <u>underlined text</u>.</p>
<button onclick="copy('demo')">Copy Keeping Format</button> 

<div id="target" contentEditable="true"></div>