Most common digital systems are non-causal. Systems that play back music or video (CD, iPod, MP3, dvd) don't just display a sound or scene based on what was stored up to that sound, they look into the "future" (digital information about sounds or scenes that are coming up) and incorporate that information into the display. This is non-causal but common. Basically
a) non-causal systems have superior performance over causal ones. For one example, one can prove that causal (physical) filters *always* introduce phase distortion, while non-causal filters can be designed with ideal properties including no phase distortion. This is important in audio, medical imaging, aerospace, etc.
b) non-causal systems can do things that physical systems cannot. Sound and image compression (mp3, jpeg, mpeg, HDTV signals, etc.) is an important example. You need to know the future signals as well as past to do a good job of compression. Error correction is another example (ever wonder why your computer reads and writes billions of bits to disk without losing data?, also how you can send voice and data over noisy and unreliable wireless cell phone links? One tradeoff is that the voice is heard with a delay on the other end), also data encryption.
On the other hand, physical systems are causal by definition--they respond to stimulus. That is true of living organisms, weather, musical instruments, things that vibrate in the wind, etc. There are also artificially designed systems that are causal, usually because a) they require instantaneous real-time response and/or b) information about the future isn't available. An example might be the control system for an active automotive suspension (ride is adjusted based on the road just driven over since you don't generally know what bumps or driver input are coming up).
You can probably think of many other examples of causal and non-causal systems.
