Decimal Numbers
The easiest way to understand bits is to compare them to something you know: digits. A digit is a single place that can hold numerical values between 0 and 9. Digits are normally combined together in groups to create larger numbers. For example, 6,357 has four digits. It is understood that in the number 6,357, the 7 is filling the "1s place," while the 5 is filling the 10s place, the 3 is filling the 100s place and the 6 is filling the 1,000s place. So you could express things this way if you wanted to be explicit:

(6 * 1000) + (3 * 100) + (5 * 10) + (7 * 1) = 6000 + 300 + 50 + 7 = 6357

Another way to express it would be to use powers of 10. Assuming that we are going to represent the concept of "raised to the power of" with the "^" symbol (so "10 squared" is written as "10^2"), another way to express it is like this:

(6 * 10^3) + (3 * 10^2) + (5 * 10^1) + (7 * 10^0) = 6000 + 300 + 50 + 7 = 6357

What you can see from this expression is that each digit is a placeholder for the next higher power of 10, starting in the first digit with 10 raised to the power of zero.

That should all feel pretty comfortable -- we work with decimal digits every day. The neat thing about number systems is that there is nothing that forces you to have 10 different values in a digit. Our base-10 number system likely grew up because we have 10 fingers, but if we happened to evolve to have eight fingers instead, we would probably have a base-8 number system. You can have base-anything number systems. In fact, there are lots of good reasons to use different bases in different situations.