When you brew or steep tea, the result is that you get a cup of tea called an infusion. You can also say you're infusing tea instead of brewing or steeping it.
Good quality tea can be brewed multiple times to yield more than one infusion. Each infusion will taste a little bit different and have different properties. If you really want to get to know a tea, sit down with it, take it through a few infusions, and notice how it evolves.
There is a Chinese saying:
"The first brew is for your enemy, the second for the servant, the third for your wife, fourth for your mistress, the fifth for your business partner and the last for yourself."
In the first infusion, the leaves are just starting to open up and release their flavor, so the flavor may be relatively weak and unimpressive. In the second infusion, the leaves have had the chance to open a little and will more readily release flavor compounds. As you go into the 3rd and 4th+ infusions you may notice different flavors pop up and fade- every tea is unique. The idea behind this quote is that the tea gets better as you continue to infuse it (and I couldn't agree more).
This infusion theory does not apply to most bagged teas. Why? Because the tea inside many bagged teas has been cut up and shredded so that it will very quickly release all its biochemical components (AKA all its flavor) and is rarely good for more than 1 infusion.
So, I recommend you do a little experiment. Have your favorite tea by your side while you work or relax and keep re-infusing it until it starts to lose flavor. Try to pinpoint where the tea hits its peak and observe how it changes. Trust me, you're in for a treat.