DiskStation Manager 4.1
Synology's firmware is called Disk Station Manager. Firmware doesn't really cover it, because it's more akin to a fully-fledged operating system and also is very different from firmware in for example routers. The AJAX-based environment works with windows and is entirely graphical, providing a nice overview of the features. There are a ton of features, too many to cover here, but a quick glance at Synology's website will give you a good idea of everything you can do with a Synology NAS.
Synology's firmware looks really good.
You are also able to install apps in DiskStation Manager 4.1. A practical interface lets you add these, including Logitech Media Server for Squeezebox, Plex Media Server, mailservers such as Zarafa, and websites such WordPress, Drupal and Joomla. It's also possible to add apps outside of the Package Center.
Package Center lets you add more features.