What is a length?  This is a deep question, addressed in great detail by euclid.  The modern interpretation of his answer is that lengths are exactly the nonnegative real numbers.  The distance on the real number line from 0 to x is identified just with the real number x.  If x,y are real numbers, then x+y is the length of the concatenation of a segment of length x with a segment of length y.

How can we find the distance between any two real numbers 0<=x<y?  Let d denote the distance from x to y.  Observe that by concatenating the segment from 0 to x to the segment from x to y, we obtain the segment from 0 to y.  Translated into symbols, this means that x+d=y.  We now do the natural algebraic manipulation, and say that the distance d from x to y is the "difference", that is, y-x.

Now, we also include the negative real numbers.  These represent imaginary lengths, which in part allow for the unification of adding and subtracting.  See the motivation for Negative Numbers.  On this two-sided real number line, given any two numbers x,y, we can find a third number x-y.  I claim that the distance between any x<y (even for negative x,y) is STILL y-x.  If instead y<x, then the distance is of course x-y.  Mathematicians have introduced a symbol to avoid the need for always dividing statements into the cases x<y and y<x.  This is the absolute value, which is defined so that |y-x| = |x-y| = whichever one of y-x and x-y is nonnegative.  Thus |x-y| is the distance from x to y.

Now let me jump ahead to R^3, the "3-d" space of vectors with three components.  We can define addition, subtraction, and the absolute value of vectors.  Addition corresponds to concatenating vectors tip-to-tail.  Subtraction is a little more complicated, but we can avoid describing it directly by first saying that -y is the vector which is of the same length and points opposite to y, and then x-y is just x+(-y).  Finally, the absolute value is now called the "norm", written ||x||, and this just means "the length of x."  Now, we can again write  ||x-y|| is the distance from x to y.

But there are many other kinds of "distance" measurements in real life.  For instance, if you want to travel from New York to Paris, the distance is not ||new york - paris||, because you can't fly through the mantle of the earth.  Instead, you must go around, and the "true" distance is the length of the great circle segment joining new york and paris.  For another example, if you want to get from the Empire State building to the Port Authority, only the crow may fly directly.  The "true" distance for you as you walk on streets and avenues is the east-west distance plus the north-south distance.

In each of these examples, we have defined a "distance function" which is different from ||x-y||.  The first takes any two points x,y on earth and gives d(x,y) = the length of the shortest great circle segment from x to y.  The second takes any two points x,y in manhattan and gives d(x,y) = the horizontal distance + the vertical distance.  These distance functions still have many properties in common with ||x-y||.  Metric space theory seeks to identify and understand these properties.  Once identified, any function satisfying these properties will be called an abstract "metric".  Metric is therefore just fancy jargon for a "distance function".

Now, let us discover the properties that should define a metric.