How to add/remove a class in JavaScript?

Here is solution for addClass, removeClass, hasClass in pure javascript solution.

Actually it's from http://jaketrent.com/post/addremove-classes-raw-javascript/

function hasClass(ele,cls) {
  return !!ele.className.match(new RegExp('(\\s|^)'+cls+'(\\s|$)'));
}

function addClass(ele,cls) {
  if (!hasClass(ele,cls)) ele.className += " "+cls;
}

function removeClass(ele,cls) {
  if (hasClass(ele,cls)) {
    var reg = new RegExp('(\\s|^)'+cls+'(\\s|$)');
    ele.className=ele.className.replace(reg,' ');
  }
}

Look at these oneliners:

1. Remove class:

   element.classList.remove('hidden');
   

2. Toggle class (adds the class if it's not already present and removes it if it is)

   element.classList.toggle('hidden');
   

That's all! I made a test - 10000 iterations. 0.8s.

I just wrote these up:

function addClass(el, classNameToAdd){
    el.className += ' ' + classNameToAdd;   
}

function removeClass(el, classNameToRemove){
    var elClass = ' ' + el.className + ' ';
    while(elClass.indexOf(' ' + classNameToRemove + ' ') !== -1){
         elClass = elClass.replace(' ' + classNameToRemove + ' ', '');
    }
    el.className = elClass;
}

I think they'll work in all browsers.

The simplest is element.classList which has remove(name), add(name), toggle(name), and contains(name) methods and is now supported by all major browsers.

For older browsers you change element.className. Here are two helper:

function addClass(element, className){
    element.className += ' ' + className;   
}

function removeClass(element, className) {
    element.className = element.className.replace(
        new RegExp('( |^)' + className + '( |$)', 'g'), ' ').trim();
}

One way to play around with classes without frameworks/libraries would be using the property Element.className, which "gets and sets the value of the class attribute of the specified element." (from the MDN documentation).
As @matías-fidemraizer already mentioned in his answer, once you get the string of classes for your element you can use any methods associated with strings to modify it.

Here's an example:
Assuming you have a div with the ID "myDiv" and that you want to add to it the class "main__section" when the user clicks on it,

window.onload = init;

function init() {
  document.getElementById("myDiv").onclick = addMyClass;
}

function addMyClass() {
  var classString = this.className; // returns the string of all the classes for myDiv
  var newClass = classString.concat(" main__section"); // Adds the class "main__section" to the string (notice the leading space)
  this.className = newClass; // sets className to the new string
}