[DanInMa]

<form id="petform" action="/" method="post">
                Enter your pets name: <input id="petinput" type="text" name="petname" />
                <input id="petnamesubmit" type="submit" value="Submit to Database" />
</form>

//   PHP MySQL: Post PetName to MySQL Database using AJAX
$(function() {
$('#petform').submit(function(){
    $.ajax({
        url: phpdb.php,
        type:'POST',
        data: $(this).serialize(),
        success: function(msg){
                alert("Data has been submitted to the Database");
            }                   
        });
    });
    });

you were getting the first error becuase you were trying to serialize the button itself becuase you binded the function to.

i put your code in a document ready section and binded directlky to the forms submit event. give this a shot