Maxwell predicted that electromagnetic waves consist of electric and magnetic fields that oscillate in phase with one another and that travel at the speed of light. For a plane EM wave:
E(x,t) = Emax cos(kx - ωt)
B(x,t) = Bmax cos(kx - ωt)
where ω = 2πf, with f the frequency in Hz, and
k = 2π/λ.
This gives ω/k = λf = c
A plane (or linearly polarized) EM wave has the electric field vectors aligned in a single plane, the magnetic field vectors aligned in a plane perpendicular to the electric field vectors, and both fields are perpendicular to the direction of propagation of the wave. EM waves are transverse waves.
Another consequence of Maxwell's equations is E/B = c.
The waves we looked at last semester needed a medium through which to travel. EM waves do not need a medium. In some sense they keep themselves going - the oscillating electric field generates an oscillating magnetic field, and the oscillating magnetic generates an oscillating electric field.