An basic example of using an upload task with a completion handler.

NSURLSession-uploadTask-example.swift

import Cocoa

let session = NSURLSession.sharedSession()
let request = NSURLRequest(URL: NSURL(string: "http://example.com")!)
let dataToUpload = "Some Arbitrary Data".dataUsingEncoding(NSUTF8StringEncoding)

let uploadTask = session.uploadTaskWithRequest(request, fromData: dataToUpload,
                                        completionHandler: { (responseData, response, error) in
    
    // Check on some response headers (if it's HTTP)
    if let httpResponse = response as? NSHTTPURLResponse {
        switch httpResponse.statusCode {
        case 200..<300:
            print("Success")
        case 400..<500:
            print("Request error")
        case 500..<600:
            print("Server error")
        case let otherCode:
            print("Other code: \(otherCode)")
        }
    }
    
    // Do something with the response data
    if let
        responseData = responseData,
        responseString = String(data: responseData, encoding: NSUTF8StringEncoding) {
            print("Server Response:")
            print(responseString)
    }
    
    // Do something with the error
    if let error = error {
        print(error.localizedDescription)
    }
})

uploadTask.resume()

Updated one:

    let session = URLSession.shared
    let request = NSURLRequest(url: NSURL(string: "http://example.com")! as URL)
    let dataToUpload = "Some Arbitrary Data".data(using: String.Encoding.utf8)
    
    let uploadTask = session.uploadTask(with: request as URLRequest, from: dataToUpload,
                                                   completionHandler: { (responseData, response, error) in
                                                    
                                                    // Check on some response headers (if it's HTTP)
                                                    if let httpResponse = response as? HTTPURLResponse {
                                                        switch httpResponse.statusCode {
                                                        case 200..<300:
                                                            print("Success")
                                                        case 400..<500:
                                                            print("Request error")
                                                        case 500..<600:
                                                            print("Server error")
                                                        case let otherCode:
                                                            print("Other code: \(otherCode)")
                                                        }
                                                    }
                                                    
                                                    // Do something with the response data
                                                    if let
                                                        responseData = responseData,
                                                        let responseString = String(data: responseData, encoding: String.Encoding.utf8) {
                                                        print("Server Response:")
                                                        print(responseString)
                                                    }
                                                    
                                                    // Do something with the error
                                                    if let error = error {
                                                        print(error.localizedDescription)
                                                    }
    })
    
    uploadTask.resume()