WhatsApp uses your phone's Internet connection (4G/3G/2G/EDGE or Wi-Fi, as available) to send and receive messages to your friends and family. You don't have to pay for every message. As long as you haven't exceeded your data limit or you're connected to a free Wi-Fi network, your carrier shouldn't charge you extra for messaging over WhatsApp.
WhatsApp Calling works in the same way. Learn more about WhatsApp Calling on: Android | iPhone
Please be aware that:
If your phone is roaming, additional mobile data charges may apply. Learn how to configure roaming settings for WhatsApp on: Android | iPhone
If you send SMS messages to your friends inviting them to use WhatsApp, service charges from your mobile provider may also apply.