In a coordinate system where all axes are orthogonal and are unit vectors (like the typical x,y,z space) the most common distance measure is the Euclidean distance, and it is a special case of the Minkowski distance. See also City block distance:

Chebychev distance is when you do the absolute value for every pair of attributes, and then only keep the one with the highest value.

Other ways to measure similarity is to measure Edit (or special case Hamming) distance, which is the number of symbol changes necessarily to transform one to the other.

We also got distances that measure angular difference, correlational distance (like Mahalanobis distance)