The basic idea of the slide rule comes from logarithms, in particular this fundamental identity: x * y = blogb(x) + logb(y). That is: adding logarithms is equivalent to multiplying numbers. The slide rule places numbers onto a ruler on a logarithmic scale; so the distance from "1" to a number "n" on the rule is the logarithm of "n". That's the whole fundamental trick to make it work.
Let's say we wanted to multiply 22.5 by 3.7. We move the center slide so that "1" on the C scale lines up with 2.25 on the "D" scale below it:
Now - since adding logarithms is multiplying numbers, and the position of a number on the C and D scales are determined by the same logarithm, that means that "3.7" on the "C" scale is in the same position as "2.25*3.7" on the D scale. So what's on the D scale at 3.7? We slide the cursor over (both to mark the position, and to make it easier to read), and find that it's at 8.3.
So the answer is 8.3 times 10 to the something. The rule doesn't tell us what. So we do it approximately in our heads. It's about 20 times 3 and a half, which is around 80. So the answer is 83. (The exact answer is 83.25, but this rule isn't big enough for us to see that.
You whipper-snappers and your fancy loggerisms! When I wuz your age, we had to do our multerplications by counting out grains of gravel! And WE LIKED IT that way!
And now you lazy kids even have Virtual Slide-Rules!!
Well, you kids just quit playing with your slide-rules on my lawn! You hear me!?
Heheheheh. This reminds me of Cliff Stoll's TED talk, where he does an neat little experiment with an o-scope and microphone to figure out the speed of sound. He's crunching the results when he runs into a division he can't do in his head, so he whips out the slide rule he carries around in his pocket and does the division on the slide rule right there!