Hi Surfer,
This is a good case of a picture being worth a thousand words. Since I don't have any pics I have to use a lot of words.
Although this post is lengthy it's the simplest way I can explain it. (Rod Machado's Private Pilot Handbook -available at Amazon - has some nice pics of this principle. Highly recommended).
The old style turn and slip indicator and the modern turn coordinator both work on the principle of gyroscopic precession. Precession is the RESULTANT force that a gyro feels when any force is applied to it. Precession is felt in the same direction as the applied force but at a position offset 90-degrees in the direction of rotation. For example, suppose you have a gyro rotating clockwise and you apply a force at the 12 o'clock position. The resultant force on the gyro will be felt at the 3 o'clock position.
In an old style turn and slip indicator the gimbal (mounting) axis is horizontal - it is connected to the gyro at the 3 o'clock and 9 o'clock positions. In order for the gyro to move (and, in turn, cause the turn and slip needle to move) a force needs to be FELT (not applied) either above or below this axis. A force FELT exactly at the 3 or 9 o'clock position would result in no movement. When the airplane turns or yaws, the force is applied at the 3 o'clock position (or 9 o'clock - depending on which way you're turning) but it is FELT at the 12 or 6 o'clock position which causes the gyro to "roll" (precess) and results in movement of the turn and slip needle. However, if the airplane ROLLS, the force is applied at the 6 or 12 o'clock position but is FELT at the 3 or 9 o'clock position. Since this is neither above nor below the gyro's "roll axis" the gyro doesn't precess and the turn and slip needle doesn't move. Thus, a turn and slip senses turn or yaw but NOT roll.
To help you visualize this, hold a quarter between your fingers with your fingers at the 3 and 9 o'clock positions. This is how the gyro on a turn and slip is mounted. As previously stated, when the airplane yaws or turns the applied force is at 3 or 9 o'clock (depending on direction) but the force is FELT at 6 or 12 o'clock. Go ahead and apply a force to the quarter at either the 6 or 12 o'clock position and notice how the quarter pivots between your fingers. This is how the gyro on a turn and slip reacts to a yaw or turn. Now imagine the airplane rolling. In this case the force applied to the gyro as the airplane rolls is at the 6 or 12 o'clock position but the force is FELT at the 3/9 o'clock position. Go ahead and apply a force at 3 or 9 o'clock and notice that the quarter does not pivot. Thus, a turn and slip cannot sense roll.
Now, let's cant the gimbal (mounting) axis a little. Change your grip on the quarter so you're holding it at the 2 o'clock and 8 o'clock positions. This is how the gyro is mounted in a modern turn coordinator. Notice that the quarter now responds to forces felt at the 3 and 9 o'clock positions in addition to forces felt at the 6 and 12 o'clock positions. Thus, a modern turn coordinator responds to turn AND roll.
I hope that was helpful.
p.s. Make sure you do this before you start flight training. After you're in flight training you may not have a quarter for the demo. Fortunately, it also works with pennies.