The spectral energy distribution of a galaxy emerges from the complex interplay of many physical ingredients, including its star formation history (SFH), metallicity evolution, and dust properties. Using Galaxpy, a new stellar population synthesis model, and SFHs predicted by the empirical model UniverseMachine and the cosmological hydrodynamical simulation IllustrisTNG, we isolate the influence of SFH on broad-band colours at z=0. By carrying out a principal component analysis, we show that physically-motivated SFH variations modify broad-band colours along a single direction in colour space: the SFH-direction. We find that the projection of a galaxy's present-day colours onto the SFH-direction is almost completely regulated by the fraction of stellar mass that the galaxy formed over the last billion years. Together with cosmic downsizing, this results in galaxies becoming redder as their host halo mass increases. We additionally study the change in galaxy colours due to variations in metallicity, dust attenuation, and nebular emission lines, finding that these properties vary broad-band colours along distinct directions in colour space relative to the SFH-direction. Finally, we show that the colours of low-redshift SDSS galaxies span an ellipsoid with significant extent along two independent dimensions, and that the SFH-direction is well-aligned with the major axis of this ellipsoid. Our analysis supports the conclusion that variations in star formation history are the dominant influence on present-day galaxy colours, and that the nature of this influence is strikingly simple.