Magento2 Notification messages

In your class, inject the Message ManagerInterface class:

__construct(\Magento\Framework\Message\ManagerInterface $messageManager) {
    $this->messageManager = $messageManager;
}

Note, Magento has deprecated methods:

$this->messageManager->addSuccess()
$this->messageManager->addError()
$this->messageManager->addWarning() 
$this->messageManager->addNotice()

So instead you should use:

$this->messageManager->addSuccessMessage()
$this->messageManager->addErrorMessage()
$this->messageManager->addWarningMessage() 
$this->messageManager->addNoticeMessage()

If you are using a controller, then most probably you have extended

\Magento\Framework\App\Action\Action

https://github.com/magento/magento2/blob/develop/lib/internal/Magento/Framework/App/Action/Action.php

This injects the \Magento\Framework\Message\ManagerInterface object in its __construct function using the \Magento\Framework\App\Action\Context $context object

So to display a message,

1. Success -

   $this->messageManager->addSuccess( __('This is your success message.') );

2. Error -

   $this->messageManager->addError( __('This is your error message.') );

3. Warning -

   $this->messageManager->addWarning( __('This is your warning message.') );

4. Notice

   $this->messageManager->addNotice( __('This is your notice message.') );

Inside other classes

class Dummy
{

    /**
     * @var \Magento\Framework\Message\ManagerInterface
     */
   private $messageManager;

   public function __construct(\Magento\Framework\Message\ManagerInterface $messageManager)
   {
       $this->messageManager = $messageManager;
   }

  public function someFunction()
  {
    $this->messageManager->addSuccess('Add your success message');
  }

}

So as far as i have noticed, Magento2 has removed session class specific notification messages.