Via the GUI

You can configure the most important settings for live or debug error handling in the presets:

Admin Tools > Settings > Configuration Presets > Debug Settings

Enable the debug settings in the Admin Tools

For more fine-grained error handling you can change various settings in:

Admin Tools > Settings > Configure Installation-Wide Options > SYS

Via configuration files

It is also possible to write changes manually into the configuration file config/system/settings.php or config/system/additional.php. Most configuration options related to error and exception handling are part of $GLOBALS['TYPO3_CONF_VARS']['SYS'].

The following configuration values are of interest:

If enabled, the login refresh is disabled and pageRenderer is set to debug mode. Furthermore the fieldname is appended to the label of fields.
If enabled, the total parse time of the page is added as HTTP response header X-TYPO3-Parsetime.
Defines a list of IP addresses which will allow development output to display. Setting to "*" will allow all. Setting it to an empty string allows none.
Configures whether PHP errors or Exceptions should be displayed.
Classname to handle PHP errors. Leave empty to disable error handling.
The E_* constants that will be handled by the error handler.
The E_* constant that will be converted into an exception by the default errorHandler.
The default exception handler displays a nice error message when something goes wrong. The error message is logged to the configured logs.
The default debug exception handler displays the complete stack trace of any encountered exception. The error message and the stack trace is logged to the configured logs.
Configures which PHP errors should be logged to the sys_log table.

Exception handler for rendering TypoScript content objects

Exceptions which occur during rendering of content objects (typically plugins) will be caught by default in production context and an error message is shown along with the rendered output. For more information and examples have a look into the TypoScript reference for config.contentObjectExceptionHandler.