Minkowski space is 4-dimensional quasi-Euclidean space (i.e., the set of points 
) together with the metric:
It is called quasi-Euclidean because the metric coefficients for space and time are different, rather than always being positive for a true Euclidean space. Whether the time coefficient or the space coefficients should be the negative ones has been debated for a century. It doesn't matter in practice. The convention above (time is positive and space negative) was the one preferred by Einstein.
A vector v in Minkowski space is said to be time-like if 
, light-like if 
and space-like if 
. The isometries of Minkowski space are the Lorentz transformations.