As of this post, this thread is 20dB (power) longer than it needed to be.
If three positive numbers P, Q and R are such that P equals Q times R (P = Q x R), then we normally say that P is R times as large as Q.
We will use alternate names for two numbers when they are used in the alternate way of describing how much larger P is than Q that is discussed below. We will refer to the number 10 as a bel, and to a number that is approximately 1.258925412 (but which will be defined precisely below) as a decibel.
In addition to saying that P is R times as large as Q, we will in another manner of speaking of relative largeness say that P is E bels times as large as Q when Q must be multiplied by a bel E times to make the answer as large as (i.e., equal to) P:
P = Q x R = Q x (bel x bel x bel x ...... x bel [E times]).
For example, P is 10 times larger than Q and is also 1 bel as large as Q if P = Q x 10 = Q x bel.
P is 100 times larger than Q and is also 2 bels as large as Q if P = Q x 100 = Q x bel x bel.
P is 1000 times larger than Q and is also 3 bels as large as Q if P = Q x 1000 = Q x bel x bel x bel. etc.
We will say that P is E bels as small as Q or that P is -E bels as large as Q if it is P (instead of Q) that must be multiplied by a bel E times in order that the answer is as large as (equal to) Q. For example, P is -3 bels as large as Q if P x bel x bel x bel = Q. Thus, it follows that if P is E bels as large as Q, then Q is -E bels as large as P.
Essentially, the answer to "how large (in bels) is P compared with Q?" is the answer to "how many times must Q be multiplied by bel (i.e.,10) so that the result is equal to P?". If P = Q, then P is 0 bels as large as Q (and of course then Q is also 0 bels as large as P), because you need no (zero) multiplications by bel to make one equal to the other.
Similarly, we will say that P is F decibels as large as Q, if Q must be multiplied by a decibel F times in order that the result is equal to P. A decibel is precisely defined by requiring that a bel is 10 decibels as large as the number one. In other words, the defining equation (in which we use the convenient abbreviation dB for decibel) is,
10 = 1 bel = 1 x (dB x dB x dB x dB x dB x dB x dB x dB x dB x dB), wherein 1 is multiplied by a decibel 10 times. This is really the same thing as saying that a decibel is the principal tenth root of a bel, but then you would be asked what an n'th root is. The decibel is a finer measure of relative largeness than the bel which is coarser.
Note that if P is E bels as large as Q, then it is also (F =) E x 10 decibels as large as Q, because of the preceding defining equation, which is why the decibel is so named. Thus, if P is 3 bels as large as Q, then P is also 30 decibels as large as Q. By pondering and using the simple rules and reasoning described above, and by practicing with actual numbers perhaps with the aid of an electronic calculator, people unfamiliar with the dB can get an intuitive feel for bel and decibel (logarithmic) measures of ratios of like quantities.
Once the person you are explaining dB to has absorbed the preceding concepts, they may come back to you at some point with a ratio P to Q that is not a whole number of bels or decibels and ask what the answer is in such a case. For example, the person may pick P = 200 and Q = 8, when P = Q x 25. This is your chance to ease them into rational exponents by pointing out that Q x bel is smaller than P, and so P is more than 1 bel as large as Q, and also Q x bel x bel is larger than P, and so P is less than 2 bels as large as Q. Next, you explain that bel x bel or 10 x 10 is written in a shorthand notation as 10^2, and 10 x 10 x 10 as 10^3, and 10 as 10^1, where 10^1 is called "10 raised to the power of 1", 10^2 is called "10 raised to the power of 2", etc. And thus 1000,000,000 is a billion (U.S.) and is also 10^9 and is also 9 bels (multiplied together). Thus the shorthand notation and the measure in bels or decibels both allow us to compactly represent giant numbers and very tiny numbers. Then show them with the y^x or 10^x function buttons on your electronic calculator that you indeed get the answer 100 when you evaluate 10^2 and the answer 1000 when you evaluate 10^3, and get the answer 10 when you evaluate 10^1 (and 1 when you evaluate 10^0). Then you explain that Q must be multiplied by 10 more than once but less than twice for the result to equal P. Next you show them by trial and error on the calculator that 10^(1.39794...) gives the answer 25. And explain that thus, in a manner of speaking, they must multiply Q by bel 1.39794... times for the result to equal P. Or P = Q x 10^(1.39794...). They can see that this is what happens on the calculator. Therefore the answer to the original question is that 200 is 25 times larger than 8, and equivalently that 200 is 1.39794... bels as large as 8. Also 300 is also 1.39794... bels as large as 12; the bels measure the ratio of P to Q, and the ratio of 200 to 8 happens to be the same as the ratio of 300 to 12. (When their understanding has advanced some more, you can show them that they can get the calculator to do the trial and error for them by finding the answer as log10(200/8) = log10 (25) = 1.39794..., as also explain the distinction between dBP and dBV).
As an aside, an explanation similar to the above can be devised for a logarithmic measure defined relative to any base (10, 2, e, ..., basically any positive non-zero number other than 1, and preferably > 1) raised to any rational exponent (such as fractions of a decibel). Real number exponents can informally be made intuitive, but a rigorous math treatment is frightfully difficult (for me). So I would steer clear of explaining real (or complex) exponents to anyone without a good background in arithmetic and elementary algebra. At the heart of the matter, exponentiation is just performing repeated multiplications while logarithms are the counting of how many multiplications are performed. And multiplication in turn is just performing repeated additions while division is the counting of how many additions are performed. And addition in its turn is just repeated counting.