If you are using ESM (ECMAScript Module) syntax like import logger from './logger'; you must specify ESM exports.
NOTE: if you are using v12, you must be using Node.js v12.17.0 or above (changelog reference) or using the --experimental-modules flag.
In addition you must specify "type": "module" in your package.json as @Evert mentioned
So:
logger.js
var url='http://mylogger.io/log';
// or export function log(message) {
function log(message){
    //send an http request;
    console.log(message);
}
export default log;
app.js
import logger from './logger';
package.json
Specify "type": "module" in your package.json
{
  "name": "<project>",
  "version": "<version>",
  "description": "<description>",
  "type": "module",
  "main": "index.js",
  "scripts": {
    "test": "echo \"Error: no test specified\" && exit 1"
  },
  ......
}
Reference Node.js ECMAScript Modules