Caffeine is a type of drug or stimulant naturally produced in the leaves of plants, like your Black Tea, it also comes from seeds and beans like coffee. Caffeine can cause mild stimulation, which increases your heart rate and alertness, the reason most people consume it.
Used in moderation caffeine is not particularly harmful. Doctors suggest that people should limit their intake to about 100 milligrams per day. To give you an idea of how much this is, here is a list of a few amounts:
Tea 70 mg
Coca-Cola 45.6 mg
As far as caffeine stunting one's growth, this is a myth. Scientists have had many concerns about this possible side effect, but thus far there has not been any research to support it. It is important to know that caffeine can have many other adverse effects on your health if consumed in higher doses. Caffeine can cause anxiety, headaches, jitters, sleep interference and contribute to some heart problems. This substance is a drug and like most stimulants it can be addictive. Once addicted, there are withdrawal symptoms to deal with such as severe headaches, temporary depression and irritability. You might have thought that caffeine was just perking you up, but it may also be playing with your attitude as well. Athletes should pay close attention to how much caffeine they consume, as well as any other person. Caffeine has a dehydrating effect on the body's cells (making them lose water), but at the same time causes your body to urinate. This two-fold mechanism causes rapid water loss from the body. It will also cause athletes to have sore muscles and delayed recovery time.
Your question is a good one. Do not stop drinking something you enjoy, just remember to consume it in moderation and be aware of other things that also contain caffeine. Drink other decaffeinated beverages to prevent dehydration also. Not everyone is affected by the same amounts, so be attuned to how your body reacts when drinking or eating substances containing caffeine.