1) Indeed, watchdog is a standard way to record own PHP errors.

2) Alternatively, if you need to immediately see error messages while debugging your Drupal pages, you may want to see them logged/printed right at the related page - in FireBug console. Sometimes is this very convenient when you can see page-related just-in-time logs. This requires - Devel module, Firebug extension to FireFox and possibly Firephp.

You can use the dfb() function to write log messages directly to the general Firebug console.

dfb($input, $label = NULL)

If you want to keep your Drupal-related log messages out of the normal Firebug console, you can write messages to the Drupal for Firebug log with the firep() function:

firep($item, $optional_title)

Drupal 8

// Logs a notice\Drupal::logger('my_module')->notice($message);// Logs an error\Drupal::logger('my_module')->error($message);

See more examples at How to Log Messages in Drupal 8.

Watchdog is the way to go for a production system no doubt but during debugging I find the drupal_set_message function useful.

It outputs the message to the screen where the 'Operation Successful'-type messages are normally displayed (so make sure you remove them before making the site Live).

http://api.drupal.org/api/function/drupal_set_message/6