Return a string based on checkbox checked

Here's an example of how you would do it with jQuery, it's just easier to do with it and since there's a jQuery tag...

Basically, you add a handler for elements that are checkboxes. Then you select only checked checkboxes with :checked and then you just access each element with the jQuery each function.

$(function(){
  $('input[type=checkbox]').on('change', function(){
    var checked = [];
    $('input[type=checkbox]:checked').each(function(index, checkbox){
      checked.push($(checkbox).attr('id'));
      });
    $('#result').text(JSON.stringify(checked, '/t'));
    });
  });

<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<table>
  <tr>
    <td>
      <label for="A">
        <input type="checkbox" name="A" id="A" value="true" />
        A
      </label>

      <label for="B">
        <input type="checkbox" name="B" id="B" value="true" />
        B
      </label>
    </td>
  </tr>
</table>
<pre id="result"></pre>

You need to check both first, since otherwise the function will return "A" and never reach the code that would return both. It is also possible to eliminate all the if-else complexity when returning from each if.

A helper function and some variables can reduce repetition and code length.

var isChecked = function(id){
     var e = document.getElementById(id);
     return (e && e.checked);
}
var getCheck = function() {
     var A = isChecked("A");
     var B = isChecked("B");
     if (A && B) return "Both";
     if (A) return "A";
     if (B) return "B";
     return "None";
}

For testing, you could add

window.onclick = function(){ console.log(getCheck()); }

demo via jsfiddle

fiddle example

The answer others provided is quite good for. And I want to make some addition:

If you want to get the value whenever user check the box, you can add a eventListen to the parent table as a delegation for the checkbox. Just as I show in the fiddle example

    <table id="mytable">
        <tr>
         <td>
          <label for="A">
           <input type="checkbox" name="A" id="A" value="true" />
            A
          </label>

          <label for="B">
           <input type="checkbox" name="B" id="B" value="true" />
            B
          </label>
        </td>
      </tr>
   </table>