What sort of things can an Arduino do?
Arduino is very flexible.  You can do a lot of neat stuff with it.  It's one of the best tools for interfacing software to the real world.  However, it might be useful to also ask "What can an Arduino not do?".  Arduino has very limited memory and I/O compared to modern computers.  So some of the things the Arduino cannot do or cannot do easily are things that require a lot of memory or access to complex peripherals, like:


video recording, processing &amp; output
high-fidelity audio recording &amp; processing
act as USB host for USB devices like flash drives, disk drives, cameras, keyboards, etc.


So, you can't easily make a video game system that hooks to your TV with an Arduino.  That doesn't mean people haven't done it, but that level of hacking is in the realm of deep voodoo, and the results still end up looking like a 1980s videogame.

Often you see Arduino boards hooked up as peripherals to a larger computer.  The computer does the A/V and the Arduino handles all the other physical world interfacing.  Another common use is the fully embedded system where a more mundane device is made "smart" with a hidden Arduino. ("Your coffee table now knows when you set cups on it and buzzes you if you don't use a coaster")  This is where Arduino seems happiest.