eq() returns the element as a jQuery object. This method constructs a new jQuery object from one element within that set and returns it. That means that you can use jQuery functions on it.
get() return a DOM element. The method retrieve the DOM elements matched by the jQuery object. But as it is a DOM element and it is not a jQuery-wrapped object. So jQuery functions can't be used.
.eq(n) retrieves the n-1th jQuery object.
.get(n) retrieves the n-1th DOM element. It'd be like doing .eq(n).
:nth-child() is for more complex selectors like :nth-child(2n+1).
You could use it in place of :eq() in selector strings, but I tend to stay away from it and use .eq() to make the selector more readable:
$('ul.parent > li.child:nth-child(2)')
$('ul.parent > li.child:eq(2)')
$('ul.parent > li.child').eq(2)