The Vonage Voice API allows developers to make worldwide outbound and inbound calls in 23 languages with various voices and accents using a virtual phone number, Ruby Gem, and a few lines of code. The tutorial covers setting up the basics, creating a Vonage Application, installing the Ruby gem, making a voice call with Ruby, and integrating the Vonage API into a Rails application to make outbound calls. The integration includes storing Call records, updating their status, and providing an NCCO to play back text to the recipient during the call.