"Matrix Loaded"
Drawing by S. Uchii, 2003 
The beginning of what Bohr calls the "rational formulation of new quantum mechanics" was provided by Heisenberg.
... the new development owas commenced inh a fundamental paper by Heisenberg, where he sycceeded in emancipating himself completely from the classical concept of motion by replaing from the vbery start the ordinary kinematical and mechanical quantities by symbols which refer directly to the individual processes demanded by the quantum postulate. This was accomplished by substituting for the Fourier development of a classical mechanical quantity a matrix scheme, the elements of which symbolize purely harmonic vibrations and are associated with the possible transitions between stationary states. (Bohr, 70-1)
But what does this mean? For a more detailed and readable exposition, the reader is referred to Dr. Tononaga's book. But here is a rough and intuitive idea.
And it should be kept in mind that matrix mechanics gives probabilities of transitions between stationary states, for example, not a unique trajectory of such transitions. One of the crucial differences between classical and matrix (quantum) mechanics appears in the commutation of the product pq. In classical mechanics,
pq = qp,
but in quantum mechanics
pq −qp = ih/2π,
and you can see Planck's constant appears in this relation (i is imaginary number). In a nutshell, this relation of commutation leads to Heisenberg's indeterminacy principle (but this was a later discovery).
Bohr, N. Atomic Theory and the Description of Nature
朝永振一郎『量子力学 I』第2版、 みすず書房、1969
See also a useful site on Microphysics, at Kyushu University: http://www2.kutl.kyushu-u.ac.jp/seminar/MicroWorld/MicroWorld.html