The Vonage Voice API allows developers to build high-quality voice applications in the Cloud, using web technologies already familiar to them. To use the API, one needs to create a Vonage application and generate public and private keys, which are used for authentication. The API provides various features such as recording and storing inbound or outbound calls, creating conference calls, sending text-to-speech messages in different languages and accents, and more. A Node.js project setup is required to interact with the API, where developers need to install the Vonage Server SDK and initialize the credentials with the application ID and private key. The `createOutboundCall` method of the Vonage Node.js library is used to make calls, specifying recipient, sender, and text-to-be-spoken information, including an NCCO object that defines how the voice should be played back. Once set up, developers can run the application using Node.js, making outbound calls with customized voices and messages.