Boolean logic is easily explained using truth tabels, but the underlying hardware is rarely mentioned in the same class. I would like to explain boolean logic from the point of view of actual transistor. My target audience is educated adults with no background in engineering, for a one-time public lecture. The goal is to have them understand why a simple cut-wire alarm circuit is designed the way it's designed, and without going "deep" into solid state physics.
You can explain them as electrically controllable switches.
- If a voltage is applied to gate and source, this means
switched on: Current may flow between drain and source. Or voltage is transferred through the transistor.
- If no voltage is applied, this means
switched off: No current may flow resp. voltage is blocked.
This may be off the mark of your question, but I have explained logic gates composed of (transistor) switches by using water pressure as analogous to electricity. Below, water will only reach the pipe-end $C$ if water pressure comes in both along pipe $A$ AND there is water pressure from $B$ that pushes up and closes the normally open switch, connecting $A$ to $C$.
If either there is no water pressure at $A$, or there is no water pressure at $B$, no water reaches $C$. So this is an AND-gate. Etc.
You could follow the suggestions made in the other posts, but I would start with an even simpler model which you could build and demonstrate easily: levers.
Consider building a contraption (a la Rube Goldberg, but much more boring) of remotely flipping a light switch (where down means light is off and up means light is on). One way to do this is to attach a long rod to the switch as follows: fix one end (use a universal joint as needed) of the rod to the switch, and fix the middle of the rod to a stationary object. You now have a logical inverter (the other end of the rod will be up when the switch is down, and down when the switch is up).
This requires a representation (0 means down and 1 means up) and a tolerance for building more complicated contraptions (concatenate two inverters to get the Boolean identity function, take two inverters and a spring or a lead weight on the end of a third inverter and make a two input AND gate by placing the three levers in a star configuration with the weight in the center; pushing on the ends on one of two of them will cause the third end to fall), but conceptually they are very simple: this only requires people having seen a lever before, and the contraptions aren't that complex. You will need very few levers to make a model of a transistor, and many people should make the analogy between lever position and voltage level easily.
Gerhard "Mixes Water And Electrons Mentally" Paseman, 2015.02.26