Newton’s theory of gravitation unified the motion of apples falling from a tree, cannonballs thrown into walls, the motion of planet earth around the sun, and even motion of all the known planetary bodies.
Electromagnetic waves gives us one understanding of X-rays, gamma radiation, visible light, infrared, radio waves, etc.
Quantum field theory unifies every microscopic theory of matter and energy we have ever known.
This list could be much longer.
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In general, see the Classification of Finite Simple Groups [0]. In terms of specific examples, the minimal classifications of knots [1], trees [2], and braids [3] are 3 instances that popped to mind. From a number-theoretic perspective, the paper Enumerating the Rationals and its derivative works are interesting reads. In terms of spatial structure and spatial decomposition, look at the classification of crystal structures [4], lattices, polytopes (zonotopes are particularly interesting), periodic and aperiodic tilings, the classification of space-filling curves, and the classification of closed surfaces [5]. From a relational perspective, look at the classification of topological spatial relations [6] and Allen's interval algebra [7]. And the one my intuition finds most intriguing is the classification of Group Theory Single Axioms and its relation to my conjecture that division is primary.
[0] Classification of Finite Simple Groups https://en.wikipedia.org/wiki/Classification_of_finite_simpl...
[1] Prime Knots https://en.wikipedia.org/wiki/List_of_prime_knots
[2] Enumerating Trees https://www.cs.virginia.edu/~lat7h/blog/posts/434.html
[3] Braid Group https://en.wikipedia.org/wiki/Braid_group
[4] Crystal Structure https://en.wikipedia.org/wiki/Crystal_structure
[5] Classification of Closed Surfaces https://en.wikipedia.org/wiki/Surface_(topology)#Classificat...
[6] Topological Spatial Relations https://en.wikipedia.org/wiki/Spatial_relation
[7] Allen's Interval Algebra https://en.wikipedia.org/wiki/Allen%27s_interval_algebra
[8] "Yet Another Paper on Group Theory Single Axioms" https://www.cs.unm.edu/~mccune/projects/gtsax/
P.S. Clifford Algebra / Geometric Algebra [00] unifies much of mathematics, e.g. in GA, Maxwell's equations can be reduced to one equation and expressed on one line [01]:
[00] Geometric Algebra https://en.wikipedia.org/wiki/Geometric_algebra
[01] Maxwell's equations formulated in terms of Geometric Algebra https://en.wikipedia.org/wiki/Mathematical_descriptions_of_t...
I think it's also interesting to distinguish between systems that reveal intrinsic order (e.g. periodic table), and systems that superimpose external order (e.g. a performance review framework).
Behavioural economics is an example of a field that desperately needs a modern-day equivalent of Mendeleev to come around and structure.
I am craving more examples of this, please, anyone, share!