jQuery, get html of a whole element

You can easily get child itself and all of its decedents (children) with Jquery's Clone() method, just

var child = $('#div div:nth-child(1)').clone();  

var child2 = $('#div div:nth-child(2)').clone();

You will get this for first query as asked in question

<div id="div1">
     <p>Some Content</p>
</div>

You can clone it to get the entire contents, like this:

var html = $("<div />").append($("#div1").clone()).html();

Or make it a plugin, most tend to call this "outerHTML", like this:

jQuery.fn.outerHTML = function() {
  return jQuery('<div />').append(this.eq(0).clone()).html();
};

Then you can just call:

var html = $("#div1").outerHTML();

You can achieve that with just one line code that simplify that:

$('#divs').get(0).outerHTML;

As simple as that.

Differences might not be meaningful in a typical use case, but using the standard DOM functionality

$("#el")[0].outerHTML

is about twice as fast as

$("<div />").append($("#el").clone()).html();

so I would go with:

/* 
 * Return outerHTML for the first element in a jQuery object,
 * or an empty string if the jQuery object is empty;  
 */
jQuery.fn.outerHTML = function() {
   return (this[0]) ? this[0].outerHTML : '';  
};