Check if array is empty or null

User JQuery is EmptyObject to check whether array is contains elements or not.

var testArray=[1,2,3,4,5];
var testArray1=[];
console.log(jQuery.isEmptyObject(testArray)); //false
console.log(jQuery.isEmptyObject(testArray1)); //true

As long as your selector is actually working, I see nothing wrong with your code that checks the length of the array. That should do what you want. There are a lot of ways to clean up your code to be simpler and more readable. Here's a cleaned up version with notes about what I cleaned up.

var album_text = [];

$("input[name='album_text[]']").each(function() {
    var value = $(this).val();
    if (value) {
        album_text.push(value);
    }
});
if (album_text.length === 0) {
    $('#error_message').html("Error");
}

else {
  //send data
}

Some notes on what you were doing and what I changed.

1. $(this) is always a valid jQuery object so there's no reason to ever check if ($(this)). It may not have any DOM objects inside it, but you can check that with $(this).length if you need to, but that is not necessary here because the .each() loop wouldn't run if there were no items so $(this) inside your .each() loop will always be something.
2. It's inefficient to use $(this) multiple times in the same function. Much better to get it once into a local variable and then use it from that local variable.
3. It's recommended to initialize arrays with [] rather than new Array().
4. if (value) when value is expected to be a string will both protect from value == null, value == undefined and value == "" so you don't have to do if (value && (value != "")). You can just do: if (value) to check for all three empty conditions.
5. if (album_text.length === 0) will tell you if the array is empty as long as it is a valid, initialized array (which it is here).

What are you trying to do with this selector $("input[name='album_text[]']")?