Adequate rest time is relative to the physical activities you undertake. Some body buildres train only one or two body parts a day, while others in the weight room might only pump iron every second day.
On the other end of the spectrum, a triathlete has a tough 6 or 7 day/week schedule of running, swimming, biking and dryland training to prepare for the grueling three-sport race of their choice.
Some athletes taper, or reduce training volume and intensity, before a big event. Some do not; training right up to and immediately after competition.
Each of these scenarios and any others you might think of still require ceratin amounts of rest or time off.
Most simply- start with enough sleep at night. 8-9 hours is plenty, and some can get by with a little bit less. Be beware of sleep deprivation, though. It leads to lack of physical performance, dulls the mental proccess and increases susceptability to injury.
The amount of time needed between your workouts depends greatly on the physical demands on the body and the level of athlete performing the training.
Overtraining means not physically resting the body and the mind as needed, leading to about the same results as lack of sleep as well as a good chance your improvment will stagnate or plateau.
If you feel you might be overtraining, get expert advice from a professional trainer in your field of training. And, at the very least, take a day or two off to enjoy the fruits of your labours.
Next: Poor diets destroy workout gains