We present the properties of a magneto-optical trap (MOT) of CaF molecules. We study the process of loading the MOT from a decelerated buffer-gas-cooled beam, and how best to slow this molecular beam in order to capture the most molecules. We determine how the number of molecules, the photon scattering rate, the oscillation frequency, damping constant, temperature, cloud size and lifetime depend on the key parameters of the MOT, especially the intensity and detuning of the main cooling laser. We compare our results to analytical and numerical models, to the properties of standard atomic MOTs, and to MOTs of SrF molecules. We load up to $2 times 10^4$ molecules, and measure a maximum scattering rate of $2.5 times 10^6$ s$^{-1}$ per molecule, a maximum oscillation frequency of 100 Hz, a maximum damping constant of 500 s$^{-1}$, and a minimum MOT rms radius of 1.5 mm. A minimum temperature of 730 $mu$K is obtained by ramping down the laser intensity to low values. The lifetime, typically about 100 ms, is consistent with a leak out of the cooling cycle with a branching ratio of about $6 times 10^{-6}$. The MOT has a capture velocity of about 11 m/s.
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