Show/Hide Field using Jquery based on user input

You could try something like:

$('#contest_max_retweet').change(
    function(){
    if ($(this).val().length) {
        $('#contest_numofwinners').hide();
    }
    else {
        $('#contest_numofwinners').show();
    }
});

JS Fiddle demo

This hides only the input element itself, not the containing li, or corresponding label, element.

Something like

$("#contest_max_retweet").change(function() {
   if($(this).val().length) {
      $("#contest_numofwinners").hide();
   }
   else {
      $("#contest_numofwinners").show();
   }
})
.trigger("change");

$('#contest_max_retweet').change(function() {
    if($('#contest_numofwinners').val() != "")
        $('#contest_numofwinners').hide();
});

The problem with change is that you have to leave the input (blur) in order for the function to fire. That's why I prefer keyup.

$("#contest_max_retweet").keyup(function() {
       if($(this).val().length > 0) {
          $("#contest_numofwinners").hide();
       }
       else {
          $("#contest_numofwinners").show();
       }
    });

I would toggle based on negation of value at change. I'd also go ahead and trigger the event handler at load to give an initial show/hide state (using a name-spaced event to avoid causing other bound functionality executing).

(Edit: I like what Brandon Boone did regarding keyup for instant action and added a comment to the event name on my code. It will, however, add a little overhead as the function is run for every stroke on the keyboard as opposed to when the field is blurred.)

Demo:

http://jsfiddle.net/JAAulde/fUKfb/3/

Code:

var namespaced_event = 'change.hideRetweet', //use `keyup.hideRetweet` here to avoid waiting for the input to be blurred
    contest_max_retweet_wrapper = $( '#contest_max_retweet' ).closest( 'li' ); //hides entire input, label, and container--could be pared down if desired

$( '#contest_numofwinners' )
  .bind( namespaced_event, function()
  {
      contest_max_retweet_wrapper.toggle( ! $( this ).val()  );
  } )
  .triggerHandler( namespaced_event );

David Thomas's answer certainly works well and I would use this for simple cases.

However, I've been working on a plugin that does this with multiple conditions and with multiple elements. I would like to share that now.

Here is the jsfiddle that works for your problem: http://jsfiddle.net/natedavisolds/rVehh/3/

Essentially, load the plugin then on load:

var React = $.extend({}, $.fn.reactor.helpers); // Easier to work with helpers
$('#contest_numofwinners_input').reactIf('#contest_max_retweet', React.IsBlank);

The nice thing is that you can chain the reactIf statement to a set of rules.

Here's the plugin.

(function($){

    $.fn.reactTo = function(selector) {
        var $elements = $(selector),
             $reactor_element = $(this),
             _proxy_event = function() {
                $reactor_element.trigger('change.reactor');
            };

        $elements.filter('select').bind('change.reactor', _proxy_event);
        $elements.filter('input').bind('keyup.reactor', _proxy_event);    

        return this;
    };

    $.fn.reactIf = function(sel, exp_func) {

        var $sel = $(sel);
        var _func = function() {
            return exp_func.apply($sel);                               
        };

        this.each(function() {
            if (!$(this).hasClass('reactor')) { $(this).reactor(); }

            var conditions_arry = $(this).data('conditions.reactor');
            if (!$.isArray(conditions_arry)) { conditions_arry = []};

            conditions_arry.push(_func);

            $(this).data('conditions.reactor', conditions_arry);
        });    

        $(this).reactTo(sel);

        return this;
    };

    $.fn.react = function() {

        this.each(function() {
           $(this).trigger('change.reactor')
        });    

        return this;
    };

    $.fn.reactor = function(options) {
        var settings = $.extend({}, $.fn.reactor.defaults, options);

        this.each(function() {
            // var opts = $.meta ? $.extend({}, settings, $this.data()) : settings;

            var $element = $(this);

            if (!$element.hasClass('reactor')) { $element.data('conditions.reactor', []).addClass('reactor'); }

            var is_reactionary = function() {  
                var conditionalArray = $(this).data('conditions.reactor');
                var r = true;

                $.each(conditionalArray, function() {
                    r = (r && this.call());
                });



                return r;                                
            }

            var reaction = function(evt) {
                evt.stopPropagation();
                if (is_reactionary.apply(this)) {
                   settings.compliant.apply($element);
                } else {
                   settings.uncompliant.apply($element);
                }
            }

            $element.bind('change.reactor', reaction);
        });

        return this;  
    };

    $.fn.reactor.defaults = {
        compliant: function() {
            $(this).show();
        },
        uncompliant: function() {
            $(this).hide();    
        }        
    };

    $.fn.reactor.helpers = {

        NotBlank: function() {
            return( $(this).val().toString() != "" )
        },

        IsBlank: function() {
            return( $(this).val().toString() == "" )
        },

        EqualTo: function(matchStr) {
            var _func = function() {
                var v = $(this).val();
                if (v) { return( v.toString() == matchStr ); }
                else { return false; }
            }
            return _func;
        },

        LessThan: function(number) {
            var _func = function() {
                var v = $(this).val();
                return(!(v && parseInt(v) > number));
            }
            return _func;   
        },

        MoreThan: function(number) {
            var _func = function() {
                var v = $(this).val();
                return(!(v && parseInt(v) < number));
            }
            return _func;  
        },

        Between: function(min, max) {
            var _func = function() {
                var v = $(this).val();
                return(!(v && (parseInt(v) > max || parseInt(v) < min)));
            }
            return _func;
        },

        BetweenSameLength: function(min, max) {
            var len = min.toString().length;
            var _func = function() {
                var v = $(this).val();
                return(!(v && v.length == len && (parseInt(v) > max || parseInt(v) < min)));
            }
            return _func;
        }
    };

})(jQuery);