Remove elements onclick (including the remove button itself) with jQuery

my suggestion would be like this.

$(function () {
    var counter = 0;

    $('a').click(function () {
        var elems = '<div>'+
              '<input type="checkbox" id="checkbox"' + (counter) + '" class="item"/>' + 
              '<input type="text" id="t1"' + (counter) + '"/>' +
              '<input type="button" class="removebtn" value="." id="removebtn"' + (counter) + '"/>' +
        '</div>';
        $('#box').append(elems);
        $("#box").append("<br />");
        $("#box").append("<br />");
        counter++;
    });

    $('.removebtn').live(function () {
        $(this).parent().remove();   
    });
});

simple demo

Suppose your remove button's id is #removebtn1234. Then

$("#removebtn1234").click(function(){
    $(this).remove();
});

should work

But for easier manipulation on multiple items, I suggest you modify to following:

$('a').click(function() {
  $('#box').append('<input type="checkbox" data-id="' + counter + '"/>');
  $('#box').append('<input type="text" data-id="' + counter + '"/>');
  $('#box').append('<input type="button" value="." class="removebtn" data-id="' + counter + '"/>');
  $("#box").append("<br /><br/>");
  counter++;
});

$(".removebtn").click(function(){
   var id = $(this).data("id");
   $("*[data-id=" + id + "]").remove();
});

Here's a solution (which seems messier than it should, but I can't place my finger on it).

$(function() {
    var counter = 0;

    $('a').click(function() {

        var checkBox = $('<input type="checkbox" id="checkbox' + (counter) + '"/>'),
            textBox = $('<input type="text" id="t1' + (counter) + '"/>'),
            removeButton = $('<input type="button" value="." id="removebtn' + (counter) + '"/>'),
            containerDiv = $("<div />");

        removeButton.click(function(e) {
            e.preventDefault();
            containerDiv.remove();
        });

        containerDiv
            .append(checkBox)
            .append(textBox)
            .append(removeButton);

        counter++;

        $("#box").append(containerDiv);
    });
});

In this solution, I make a variable of the jQuery reference. This way we can call things like checkBox.remove() and it will reference only that checkbox (without trying to work out the URL).

I have modified it a bit and removed the <br /> tags and wrapped everything in a <div />. This way you can just remove the container div and all the elements will go.

Example: http://jsfiddle.net/jonathon/YuyPB/