Would I need to use a RTC to have the lights run on a 16 hour on/8 hour off cycle? The timing doesn't need to be super accurate.
Problems might start though once the micro controller is reset or power is lost for a short while.
Attached to an PC a standard arduino will be already be reset when the PC is started and... each time you start communicating with the arduino.
When the controller would reset after 15 hours light for some reason, the arduino may see it as the beginning of a new day and power the lights on for another 16 hours.
You may be able to store the time in eeprom every minute or so to check what arduino was doing once it restarts. That... won't unfortunately tell you how long the arduino has been off.
Should it restart after 8 hours in the above example, the arduino won't know, it turns the sun on for an hour and turns it off after that for 8.
Loads of other applications you can simply turn on/off when you need them, a living stock (plants/fish in aquarium etc) usually needs a pretty accurate 24hour day. A few minutes inaccuracy each day probably won't matter.
If it's a lot more fish and plants may start behaving quite strange though...