An example error object returned in a response body looks like this: 
{
  "errors": [
    {
      "status": "400",
      "title": "Bad Request",
      "detail": "The request could not be understood by the server due to malformed syntax.",
      "source": {
        "pointer": "/data/attributes/first-name"
      }
    }
  ]
}
Other potential error codes that may be returned are:

401 Unauthorized

403 Forbidden

In all cases the response will be a JSON object with a single key errors which will be an array of error objects. For more information, check out the responses in the API reference.