A ratio is a sort of mathematical metaphor, an analogy used to compare different amounts of the same measure. You could almost consider any type of measurement a ratio, since every measurement in the world has to have some sort of reference point. This fact alone makes measurement by ratio one of the most basic of all forms of quantification.
Units of Measure
A ratio compares two things in the same unit of measure. It doesn't matter what that unit of measure is -- pounds, cubic centimeters, gallons, newton-meters -- it matters only that the two are measured in the same units. For instance, you can't compare 1 part fuel to 14 parts of air if you're measuring fuel in pounds and air in cubic feet.
Modes of Expression
You can express a ratio either in narrative form or in symbolic mathematical notation. You can express ratio as "the ratio of A to B," "A is to B," "A : B" or the quotient of A divided by B. For example, you can express a ratio of 1 to 4 as 1:4 or 0.25 (1 divided by 4).
Equality of Ratios
You can use ratios as direct analogies to compare one thing to another, notating it either with an "=" sign or verbally. For instance, you can say "A is to B as C is to D," or you can say, "A:B = C:D." In this instance, A and D are the "extremes" and B and C are called the "means." For example, you can say, "1 is to 4 as 3 is to 12," or you can say "1:4 = 3:12."
Ratios as Fractions
In practice, ratios act something like fractions. You can replace the colon with a division sign and still arrive at the same result. As in the previous example, 1/4 (1 divided by 4) and 3/12 (3 divided by 12) both come out to 0.25. This is consistent with the last mode of expression. So any ratio may be expressed as A divided by B.
Any series of three or more ratios can string together to create a continued or serial proportion. As an example, "1 is to 4 as 3 is to 12 as 4 is to 16" and "1:4 = 3:12 = 4:16" are both continued proportions. Expressing them as decimal figures (dividing the first number by the second in each proportion), you indeed find that 0.25 = 0.25 = 0.25.