What is meant by "input/output impedence" of a transistor stage in amplifiers?
Input impedance in this context refers to "small-signal input impedance". Before I offer an intuitive explanation of what that is I will explain what it is NOT:

It is NOT:
1. Put a battery across it.
2. Look how much current flows.
3. Call the ratio of the voltage to the current the impedance.

Again it is NOT this. To see why, consider a simple example. What is the input impedance of a capacitor? If you put a battery across it and tried to measure the current, you'd measure infinity. But of course we know that we characterize a capacitor as having an input impedance of 1/(jwC). If you don't understand this last bit, don't worry.

Small-signal input impedance is this.

You WIGGLE a node voltage up, and you watch how large the corresponding WIGGLE in the current that rushes away you is.

I like water analogies and despite the fact that they seem silly I think they're a very useful first step in developing intuition for electronics.

Consider a long hose with a water balloon at either end. A red balloon on one end and a blue balloon on the other. Suppose that this system is filled with water.

Imagine you give the red balloon a light squeeze, the blue balloon will get bigger as water is displaced from the red balloon, through the hose, into the blue balloon.

The ratio of how hard you squeezed to the amount of water wooshing the hose as you squeeze is the impedance! Notice that, like a capacitor, an impedance is defined even though no "DC" water flows.

Now make the hose thinner and apply the same pressure as before. What happens? You squeeze but less water rushes away because the hose resists. 

When I look at circuits I imagine myself standing on the node, bending down, yanking it up, and watching how much current "rushes away" from me. If it's a lot of current, I know that node has a low impedance. If it's a little bit of current, that node has a high impedance.

Combine this intuitive explanation with the commonly available definitions given in textbooks which are more mathematical in nature and you'll be on your way.