A Discord bot that can send and receive SMS messages, including notifications for important events such as members joining or leaving the server and mentions, can be built using Twilio Programmable Messaging and Node.js. The bot requires a Discord account, a Twilio account with a phone number, and Node.js v16+ installed on the machine. The bot is set up by creating a new Discord application, enabling the necessary intents, giving it permissions to join the server, and setting up an Express server to handle incoming SMS messages from the Twilio number. The bot can be tested using ngrok and configured to work with Twilio's Messaging settings.