When you react to something, you respond to it. If someone pokes you in class, you might react by sticking out your tongue at them. The teacher might then react by giving you both detention.
If someone says to you, "act, don't react," they are telling you not to automatically respond to something that someone does. Instead, you should think about it first and then decide what you want to do. In science, the word react is used to describe what happens when two substances combine and cause a chemical change in each other. When you add baking soda to vinegar, the chemicals in each react and create a foamy lava like substance.