You should certainly check out guides for your specific class to ensure you're using the right method, whether that's clipping - replacing a DoT before it's expired - or allowing something like a mage Living Bomb to explode. To maximise DoT damage, efficient re-casting is key.
This is important to ensure DoT classes are maximising their DPS, and yes, the Blizzard standard UI does represent them, but the time display is not that clear. What they do is provide an easy-to-read representation of what DoTs you have on a given target, be it your focus, your target, or any target in the area, depending on your choice of addon. If you're playing a class whose damage relies on DoTs, and there are plenty, you should definitely consider using one of these addons.
DoT trackers are addons which monitor your DoTs, or Damage over Time effects.
