How can I add a p tag in between id=p1 and id=p3 when I click the button "Add P2"?
Here is the code I have so far.
function addParagraphs()
{
    var p2 = "<p>paragraph 2</p>";
    document.getElementsById("p1").appendChild(p2);
}
<p id ="p1" > paragraph 1</p>
<p id ="p3" > paragraph 3</p>
<p id ="p4" > paragraph 4</p>
<input id="b2" type='button' onclick='addParagraphs()' value='Add P2'/>
The simplest solution is to use the insertAdjacentHTML method which is very convenient for manipulations with HTML strings:
function addParagraphs() {
    var p2 = "<p>paragraph 2</p>";
    document.getElementById("p3").insertAdjacentHTML('beforebegin', p2);
}
<p id="p1">paragraph 1</p>
<p id="p3">paragraph 3</p>
<p id="p4">paragraph 4</p>
<input id="b2" type='button' onclick='addParagraphs()' value='Add P2' />
Another option is to use the insertBefore method, but it's a bit more verbose:
function addParagraphs() {
    var p2 = document.createElement('p');
    p2.innerHTML = 'paragraph 2';
    var p3 = document.getElementById("p3");
    p3.parentNode.insertBefore(p2, p3);
}
<p id="p1">paragraph 1</p>
<p id="p3">paragraph 3</p>
<p id="p4">paragraph 4</p>
<input id="b2" type='button' onclick='addParagraphs()' value='Add P2' />
You have a typo. It's getElementById, not getElementsById. Only one single element can be gotten with a certain ID.
And you should be using createElement and insertBefore to insert the element between the two other elements, appendChild appends to the element.
function addParagraphs() {
    var p2 = document.createElement('p');
    p2.innerHTML = "paragraph 2";
    var elem = document.getElementById("p1");
    elem.parentNode.insertBefore(p2, elem.nextSibling);
}
<p id ="p1" > paragraph 1</p>
<p id ="p3" > paragraph 3</p>
<p id ="p4" > paragraph 4</p>
<input id="b2" type='button' onclick='addParagraphs()' value='Add P2'/>
A simple jQuery solution would be to use .insertAfter:
function addParagraphs() {
    var p2 = "<p>paragraph 2</p>";
    $(p2).insertAfter("#p1");
}
JSFiddle
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