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We propose several descriptive measures to characterize the arrangements of planetary masses, periods, and mutual inclinations within exoplanetary systems. These measures are based in complexity theory and capture the global, system-level trends of each architecture. Our approach considers all planets in a system simultaneously, facilitating both intra-system and inter-system analysis. We find that based on these measures, Kepler's high-multiplicity ($N\geq3$) systems can be explained if most systems belong to a single intrinsic population, with a subset of high-multiplicity systems ($\sim20\%$) hosting additional, undetected planets intermediate in period between the known planets. We confirm prior findings that planets within a system tend to be roughly the same size and approximately coplanar. We find that forward modeling has not yet reproduced the high degree of spacing similarity (in log-period) actually seen in the Kepler data. Although our classification scheme was developed using compact Kepler multis as a test sample, our methods can be immediately applied to any other population of exoplanetary systems. We apply this classification scheme to (1) quantify the similarity between systems, (2) resolve observational biases from physical trends, and (3) identify which systems to search for additional planets and where to look for these planets.