meaning of metric vs. statistic vs. parameter From early on in stats education, you get the difference between statistics and parameters drilled into your head.  However, I've more recently (outside of school) encountered the term "metric," and I've begun using it a lot myself. (e.g. talking about developing a metric for gauging timely processing)
my actual question:
What is the appropriate way to use the term "metric," and how is its meaning different from "statistic" and different from "parameter"?
 A: As already mentioned by Kodiologist, metric is rather informal name for e way of measuring something, e.g. this is how it is used by Google Analytics

Metrics are quantitative measurements. The metric Sessions is the
  total number of sessions. The metric Pages/Session is the average
  number of pages viewed per session.

In mathematics, metric is a function that measures distance between two points.
The first, informal, definition of metric is consistent with the definition statistic, i.e. function of a sample. On another hand, if you are estimating some quantity of underlying probability distribution (the parameter, see also: Is any quantitative property of the population a "parameter"?), then the function is called estimator.
A: This sense of the word "metric" is informal. It means a way to measure or quantify something.
There's an unrelated formal sense of the word "metric" that arises in analysis. There, a metric is a real-valued binary function that is nonnegative, returns 0 iff its arguments are equal, is symmetric, and satisfies the triangle inequality.
A: What is the appropriate way to use the term "metric," and how is its meaning different from "statistic" and different from "parameter"?
... was the question first asked. It has not been answered. 
I believe the term parameter is the original, traditional expression used to describe a range of defining limits, and also used to describe a state (escalating or declining) of progress or performance. 
The term metric is an Americanisation originally unfamiliar to other English-speaking people. The use of American neologisms and idiom has grown throughout the world with the expansion of the American internet and media. So too, the term metrics has come to replace parameters. Each can be used interchangeably, however UK residents would recognise parameters more readily.
A: A metric is a measurement and therefore has units: a parameter may not. For example a mole in chemistry is the amount if stuff that has 6*10^23 molecules is atoms or whatever, and is a dimensionless number - a parameter,  not a metric - that has no units.
A metric space is any set of data where distance between two points is the same in both directions and concepts of straight and point make sense. (There is a very clear definition that is worth looking up if you're interested.)
Other definitions are available.
