As I read it, you want:

• one <div class="names"> per todo (in this case inside .contents) surrounding the todo text in a span and the buttons "Edit" / "Remove"
• on clicking "Edit", the span is exchanged for an input field and an "Update" button, "Edit" and "Remove" are removed.
• on clicking "Update", the prior state with the new text is recreated (span, Edit, Remove)

After you follow Inkbug`s advice from his comment, you should probably do this:

• cache the todo root element (var todoRoot = $(...);)

• rename addUpdateField/.update to edit/.edit and updateBtn to update - consistent with the button texts and easier to read

• change the api of your todo.js functions. For update / edit / remove, just use the dom node of the div as the argument:

   function(target, action) {
       todoRoot.on('click', target, function(e) {
           e.preventDefault();
           e.stopPropagation();
           action.call(todo, $(this).parents('.names').first());
       });
   }
  
   bindClick('.remove', todo.remove);
   bindClick('.edit',   todo.edit);
   bindClick('div',     todo.update);
  

• don't use that many appends. Put it all in a string (that can contain more than one DOM node!)

• don't create / delete all the nodes. Create all three buttons, use .toggleClass(...) and CSS to make all buttons/fields disappear that you don't need. That moves all your HTML into the add function and goes easy on the DOM

• move the bindings into the Todo function. You don't want to have to remember to bind these events for every Todo list you create

As I read it, you want:

• one <div class="names"> per todo (in this case inside .contents) surrounding the todo text in a span and the buttons "Edit" / "Remove"
• on clicking "Edit", the span is exchanged for an input field and an "Update" button, "Edit" and "Remove" are removed.
• on clicking "Update", the prior state with the new text is recreated (span, Edit, Remove)

After you follow Inkbug`s advice from his comment, you should probably do this:

• cache the todo root element (var todoRoot = $(...);)

• rename addUpdateField/.update to edit/.edit and updateBtn to update - consistent with the button texts and easier to read

• change the api of your todo.js functions. For update / edit / remove, just use the dom node of the div as the argument:

   function(target, action) {
       todoRoot.on('click', target, function(e) {
           e.preventDefault();
           e.stopPropagation();
           action.call(todo, $(this).parents('.names').first());
       });
   }
  
   bindClick('.remove', todo.remove);
   bindClick('.edit',   todo.edit);
   bindClick('div',     todo.update);
  

• don't use that many appends. Put it all in a string (that can contain more than one DOM node!)

• don't create / delete all the nodes. Create all three buttons, use .toggleClass(...) and CSS to make all buttons/fields disappear that you don't need. That moves all your HTML into the add function and goes easy on the DOM

• move the bindings into the Todo function. You don't want to have to remember to bind these events for every Todo list you create

As I read it, you want:

• one <div class="names"> per todo (in this case inside .contents) surrounding the todo text in a span and the buttons "Edit" / "Remove"
• on clicking "Edit", the span is exchanged for an input field and an "Update" button, "Edit" and "Remove" are removed.
• on clicking "Update", the prior state with the new text is recreated (span, Edit, Remove)

After you follow Inkbug`s advice from his comment, you should probably do this:

• cache the todo root element (var todoRoot = $(...);

• rename addUpdateField/.update to edit/.edit and updateBtn to update - consistent with the button texts and easier to read

• change the api of your todo.js functions. For update / edit / remove, exprect the dom node of the div:

   function(target, action) {
       todoRoot.on('click', target, function(e) {
           e.preventDefault();
           e.stopPropagation();
           action.call(todo, $(this).parents('.names').first());
       });
   }
  
   bindClick('.remove', todo.remove);
   bindClick('.edit',   todo.edit);
   bindClick('div',     todo.update);
  

• don't use that many appends. Put it all in a string (that can contain more than one DOM node!)

• don't create / delete all the nodes. Create all three buttons, use .toggleClass(...) and CSS to make all buttons/fields disappear that you don't need. That moves all your HTML into the add function and goes easy on the DOM

• move the bindings into the Todo function. You don't want to have to remember to bind these events for every Todo list you create

As I read it, you want:

• one <div class="names"> per todo (in this case inside .contents) surrounding the todo text in a span and the buttons "Edit" / "Remove"
• on clicking "Edit", the span is exchanged for an input field and an "Update" button, "Edit" and "Remove" are removed.
• on clicking "Update", the prior state with the new text is recreated (span, Edit, Remove)

After you follow Inkbug`s advice from his comment, you should probably do this:

• cache the todo root element (var todoRoot = $(...);)

• rename addUpdateField/.update to edit/.edit and updateBtn to update - consistent with the button texts and easier to read

• change the api of your todo.js functions. For update / edit / remove, just use the dom node of the div as the argument:

   function(target, action) {
       todoRoot.on('click', target, function(e) {
           e.preventDefault();
           e.stopPropagation();
           action.call(todo, $(this).parents('.names').first());
       });
   }
  
   bindClick('.remove', todo.remove);
   bindClick('.edit',   todo.edit);
   bindClick('div',     todo.update);
  

• don't use that many appends. Put it all in a string (that can contain more than one DOM node!)

• don't create / delete all the nodes. Create all three buttons, use .toggleClass(...) and CSS to make all buttons/fields disappear that you don't need. That moves all your HTML into the add function and goes easy on the DOM

• move the bindings into the Todo function. You don't want to have to remember to bind these events for every Todo list you create