Code.gs

//  1. Enter sheet name where data is to be written below
        var SHEET_NAME = "DATA";
         
//  2. Run > setup
//
//  3. Publish > Deploy as web app 
//    - enter Project Version name and click 'Save New Version' 
//    - set security level and enable service (most likely execute as 'me' and access 'anyone, even anonymously) 
//
//  4. Copy the 'Current web app URL' and post this in your form/script action 
//
//  5. Insert column names on your destination sheet matching the parameter names of the data you are passing in (exactly matching case)
 
var SCRIPT_PROP = PropertiesService.getScriptProperties(); // new property service
 
// If you don't want to expose either GET or POST methods you can comment out the appropriate function
function doGet(e){
  return handleResponse(e);
}
 
function doPost(e){
  return handleResponse(e);
}
 
function handleResponse(e) {
  // shortly after my original solution Google announced the LockService[1]
  // this prevents concurrent access overwritting data
  // [1] http://googleappsdeveloper.blogspot.co.uk/2011/10/concurrency-and-google-apps-script.html
  // we want a public lock, one that locks for all invocations
  var lock = LockService.getPublicLock();
  lock.waitLock(30000);  // wait 30 seconds before conceding defeat.
   
  try {
    // next set where we write the data - you could write to multiple/alternate destinations
    var doc = SpreadsheetApp.openById(SCRIPT_PROP.getProperty("key"));
    var sheet = doc.getSheetByName(SHEET_NAME);
     
    // we'll assume header is in row 1 but you can override with header_row in GET/POST data
    //var headRow = e.parameter.header_row || 1; Hawksey's code parsed parameter data
    var postData = e.postData.contents; //my code uses postData instead
    var data = JSON.parse(postData); //parse the postData from JSON
    var headers = sheet.getRange(1, 1, 1, sheet.getLastColumn()).getValues()[0];
    var nextRow = sheet.getLastRow()+1; // get next row
    var row = []; 
    // loop through the header columns
    for (i in headers){
      if (headers[i] == "Timestamp"){ // special case if you include a 'Timestamp' column
        row.push(new Date());
      } else { // else use header name to get data
        row.push(data[headers[i]]);
      }
    }
    // more efficient to set values as [][] array than individually
    sheet.getRange(nextRow, 1, 1, row.length).setValues([row]);
    // return json success results
    return ContentService
          .createTextOutput(JSON.stringify({"result":"success", "row": nextRow}))
          .setMimeType(ContentService.MimeType.JSON);
  } catch(e){
    // if error return this
    return ContentService
          .createTextOutput(JSON.stringify({"result":"error", "error": e}))
          .setMimeType(ContentService.MimeType.JSON);
  } finally { //release lock
    lock.releaseLock();
  }
}
 
function setup() {
    var doc = SpreadsheetApp.getActiveSpreadsheet();
    SCRIPT_PROP.setProperty("key", doc.getId());
}