Step 1: Explain what they are.
An electromagnetic wave is a self-sustaining disturbance made of coupled oscillating electric and magnetic fields. A changing electric field generates a magnetic field and a changing magnetic field regenerates the electric field, so the pair keeps each other going and marches forward through space, even where there is no matter.
Step 2: State their nature.
Because both fields point at right angles to the travel direction, the wave is transverse. In vacuum every such wave moves at \(c = 1/\sqrt{\mu_0\varepsilon_0} = 3\times10^{8}\) m/s, and the field magnitudes obey \(E_0/B_0 = c\), the two fields peaking together (in phase).
Step 3: Picture the wave.
Let the direction of propagation be the x-axis. Draw a sinusoidal curve for \(\vec{E}\) oscillating along the y-axis, its crest labelled with amplitude \(E_0\). Alongside it, in the perpendicular x-z plane, draw a second sinusoid for \(\vec{B}\) oscillating along the z-axis, its crest labelled \(B_0\). Both curves rise and fall together along x, and \(\vec{E}\times\vec{B}\) gives the forward direction of the wave.
\[\boxed{E,\ B,\ \text{and propagation are mutually perpendicular; } c = E_0/B_0}\]