The dynamics which govern the establishment of pattern and form in multicellular organisms remain a key problem of developmental biology. We study this question in the case of morphogenesis during aggregation of the slime mould Dictyostelium discoideum. Here detailed experimental information allows the formulation of a mechanistic model in which the central element is the coupling of the previously much-studied intracellular cyclic AMP signalling with the chemotactic cell response in cyclic AMP gradients. Numerical simulations of the model show quantitatively how signal relay, chemotactic movement and adaptation orchestrate the collective modes of cell signalling and migration in the aggregating cell layer. The interaction of chemotaxis with the cyclic AMP excitation waves causes the initially homogeneous cell layer to become unstable towards the formation of a branching cell stream pattern with close cell-cell contacts as observed in situ. The evolving cell morphology in turn leads to a pattern of non-homogeneous excitability of the medium and thus feeds back into the cAMP dynamics. This feedback can explain the decrease in signalling period and propagation speed with time, as well as observations on the structure of the spiral wave core in this self-organized excitable medium.