In addition to WhatsApp (read above), Facebook Messenger and Hangouts are some other apps to consider.
Facebook Messenger is more feature-rich, but doesn't have as many privacy and security settings.
The messaging app by WhatsApp's parent company, Facebook Messenger (free, iOS, Android, the web), has some pretty killer features, like being able to use high-definition video and audio calling on mobile or web. Messenger is unique because you can send money directly through the app in the US. There are also bots built into Messenger that can help you diagnose that weird rash or shop for you. One thing to note: users know when you've read their messages (and vice versa) and there's no straightforward way to disable read receipts, sadly.
The app recently rolled out a new, fully encrypted feature called "Secret Conversations," which ensures that the message's content can't be read by law enforcement or the company itself. The reason why Messenger is only for the ~moderately paranoid~ is because the encryption feature is opt-in, and needs to be turned on for every conversation, unlike WhatsApp, which automatically encrypts every chat by default. Additionally, "Secret Conversations" only encrypts text messages, photos, and videos sent in the thread, but it doesn't protect audio and video calls.
Google Hangouts is fine, but isn't as secure.
Hangouts (free, iOS, Android, and web) puts text messaging, audio calling, and video calling in one place – but it does not offer full encryption, so Google can wiretap conversations at the request of law enforcement. You'll need to use Google Allo's incognito mode for messaging and Google Duo for video chatting with end-to-end encryption.
And unlike WhatsApp or Facebook Messenger, which allow you to sign up with just your phone number and without a Facebook account, Hangouts requires a Google account.