The Call the Milk Wins Netbook Week is a competition where developers can showcase their skills by integrating the Twilio API with the popular Remember the Milk service. To participate, developers need to apply for an RTM api key and provide their phone number, then call a specific number to add a new task to their list. The week's challenge is to open source their code, allowing others to build upon it. The competition encourages developers to contribute to the open source community using Twilio, with resources available on the Twilio website, including API reference documentation, SDKs, and tutorials.