How can I modify the WordPress default widget output?

I don’t want to style the default widget with only CSS. I want to display the default ‘Categories’ widget content with my own HTML structure.

Is there available any filter or hook to do that?

To expand on Mark’s answer, there’s not much (generally) available in the way of filters in the default WordPress widgets (except for perhaps widget_text).

But adding your own custom widget is easy – put this in your functions.php:

require_once("my_widget.php");
add_action("widgets_init", "my_custom_widgets_init");

function my_custom_widgets_init(){
  register_widget("My_Custom_Widget_Class");
}

Then you simply want to copy the existing categories widget from wp-includes/widgets/class-wp-widget-categories.php to my_widget.php in your theme, and change the class name to the same name as that used in the call to register_widget() above.

Then make whatever changes you like! I suggest changing the title too so you can distinguish it from the default Categories widget.