A radio signal, a beam of sunlight, and the X-ray that images a broken bone are the same physical phenomenon at different scales. All three are electromagnetic waves — self-sustaining ripples of electric and magnetic fields that travel through space, carrying energy from one place to another without needing to carry any matter along with them.

Start with the two fields separately, since you've likely met them before in isolation. An electric field is the invisible influence around any charged particle that pushes or pulls other charges — it's what makes a balloon stick to your hair after you rub it on a sweater. A magnetic field is the invisible influence around a moving charge or a magnet that pushes or pulls other moving charges — it's what makes a compass needle point north. For most of the 1800s, physicists treated electricity and magnetism as related but separate topics. Then James Clerk Maxwell showed, in a set of equations now called Maxwell's equations, that they aren't separate at all: a changing electric field creates a magnetic field, and a changing magnetic field creates an electric field. Each one regenerates the other.

That mutual regeneration is the key to the whole spectrum. Picture an electric field that oscillates — grows, shrinks, reverses, grows again. As it changes, Maxwell's equations say it must generate a magnetic field, and because the electric field keeps changing, the magnetic field it produces keeps changing too. But a changing magnetic field generates its own electric field in turn. The two fields leapfrog each other outward, each one's change fueling the next, forming a traveling disturbance that needs no wire, no water, no air — nothing but empty space — to move through. That self-sustaining leapfrog is an electromagnetic wave.