To receive a POST request with server-side Swift using Vapor, you need to create a Codable struct to map the data from the request body and then use this struct in your route. After setting up your project with Swift Package Manager and Vapor, you can add routes for GET, POST, and other types of requests. To handle incoming POST requests, you'll need to decode the request body using your Codable struct. You can test your application by running `curl` commands or using a tool like ngrok to expose your local server to the internet. Once set up, you can use Vapor to create powerful backend APIs for your iOS apps and other applications.