Brandon Sherman created an app that sends business cards over SMS using Twilio and Node.js, after realizing he didn't have any with him at a conference. The app listens for keywords sent via SMS and responds with his contact information, allowing him to share selective information with people. To build this app, one needs a Twilio account and phone number, and the Node.js code creates a TwiML MessagingResponse object based on the received keyword. The app was extended at DerbyCon to include tracking who reached out, and Brandon encourages users to explore more possibilities with Twilio's APIs, such as building smart response bots or custom group messaging services.