metric spaces

Have you tried wikipedia, or google?
A metric is just another term for a standard measurement or "distance generator". A space with a metric, for instance, is ordinary 3-space with distance and time "standards", such as metres per second.

Completion means all distances are exact and the space is then closed, under measurement.
Metrics in a computer space are "capacitance", "current" etc that become "bits", so the metric of digital computer space is the bit, and bits are closed by measurement (33% = 0, 67% = 1).

Suppose your metric space is the interior of a circle in 2 dimensional Euclidean space. Let's say it does not include the perimeter of the circle. You could have a sequence of points that approached a point on the perimeter. It would be approaching a limit that was not a point in the metric space, so the metric space is not "complete".

To be more specific, suppose that we let the space be the interior of the circle with center (0,0) and radius (1,0). The sequence of points: (0,0), (0,1/2), (0,3/4), (0,7/8), (0,15/16).... does not converge to any point inside the circle. Yet the sequence looks like it "ought" to converge since the points are getting closer and closer to each other. The precise mathematical definition of a sequence that "ought" to converge is given by stating the conditions for a "Cauchy" sequence. To say that a metric space is "complete" means that all sequences that "ought" to converge do converge to a point inside the metric space.

There is good reason to treat the idea of metric space abstractly. Some of the most interesting metric spaces are infinite dimensional. For example, consider functions f(x) that can be specified by an infinite number of coordinates ( c0, c1, c2,.....) where the c's are the coefficients of its expansion in a Taylor series about x = 0. Are there sets of such functions where a metric can be defined? Are there sets of such functions that form a complete metric space?

So studying metric spaces is more than simply studying n-dimensional Euclidean space in an abstract way.