My experience was that I started feeling generally tired and couldn't raise my heart rate to its normal levels. I took a day off to give my legs chance to recover, and the next day I felt great, and my heart rate was back up where it should be.
That's not really overtraining though - one day off doesn't cure chronic overtraining. There's a big difference between overtraining and just being a bit tired after a few heavy days cycling.
The question wasn't about chronic overtraining. The OP just asked what the signs are of "overtraining". In my experience, as a long time runner as well as a fairly novice cyclist, lacking strength in your working muscles is the first sign that you're doing too much, and that your efforts aren't resulting in improved fitness levels. Taking a rest at that point prevents getting into a situation of chronic overtraining.
To actually become overtrained usually takes more will power than most people have, most people will naturally stop or back off before they actually become chronically over trained, only those with a strong will to succeed will drive on relentlessly when faced with such symptoms.
Yep, overtraining isn't the same as overdoing it, the latter can be recovered fairly easily and you can continue cycling at the same time, just by being more sedate. Though any injuries you may have picked up won't be helped. The former is a very different kettle of fish and I've seen claims that it's more about an addictive personality than a strong will. As in a truly strong will would be strong enough to back down when need be, whereas a helpless addict just carries on.
I think the addict comparison is most likely another valid case, maybe even a more common scenario.
Signs of actual over training (not just being tired) include, difficulty sleeping, low mood, changes in appetite, lack of motivation, frequently occuring illnesses due to a lull in your immune system, poor performance, elevated resting heart rate over several days, etc.