When talking about light or radio waves, you're probably talking about Classical Electrodynamics, which does not include the notion of a photon. In Classical EM, light is an EM wave in the sense that if you look at the electric field E or magnetic field B (of a plane wave) at a fixed point in space it is varying sinusoidally. So the waviness is in the amplitude of the E and B fields.
Once you talk about photons, you're in the realm of Quantum Mechanics (QM), and yes things are harder to understand.
It's actually all just fields according to the Standard Model (particle physics), a quantum field theory (QFT).
In QFT there's a field for each fundamental particle that permeates the whole universe. E.g. an electron field, a photon field, etc. Disturbances in these fields are what one would call particles in non-relativistic QM.
So Classical -> QM (quantum system, classical observer/apparatus) -> QFT (quantum everything)
In classical EM, light is a wave. In QM, light is particles. In QFT particles are just disturbances in the all-pervading fields.
Binney has said that QM is just measurement for grownups, or some such. What is a measurement? It's when the system you're observing becomes entangled with the measuring device. We don't know the exact state of every atom in our measuring device, but these could all perturb the system we're measuring. So QM is a hack where you treat the system as quantum but the observer/measuring device as classical which is why you need this confusing wave-function collapse. It was a conscious choice in the development of the theory. This last bit might give some insight into why trying to sense the photon at one of the double-slits ruins the interference pattern.