Short answers, pulled from the story.
A wave is a propagating dynamic disturbance, meaning a change from equilibrium, in one or more quantities. The two types most commonly studied in classical physics are mechanical waves, such as sound and seismic waves, and electromagnetic waves, such as light.
A traveling wave is one in which the entire waveform moves in a single direction. A standing wave forms when two identical periodic waves travel in opposite directions, producing an envelope that remains in a constant position with nodes where the amplitude falls to zero.
In a transverse wave the field disturbance at each point is described by a vector perpendicular to the direction of propagation, while in a longitudinal wave those vectors align with the propagation direction. Electromagnetic plane waves are strictly transverse, and sound waves in fluids such as air can only be longitudinal.
Both electromagnetic waves and gravitational waves propagate at the speed of light. James Clerk Maxwell showed in the 19th century that the electric and magnetic fields satisfy the wave equation at the speed of light, and gravitational waves produced by the mutual rotation of binary stars travel at the same speed.
The first observation of gravitational waves was announced on the 11th of February 2016. Gravitational waves are disturbances in the curvature of spacetime, predicted by Einstein's theory of general relativity.
The Doppler effect is the change in frequency of a wave in relation to an observer who is moving relative to the wave source. It is named after the Austrian physicist Christian Doppler, who described the phenomenon in 1842.
Louis de Broglie postulated that all particles with momentum have a wavelength equal to the Planck constant divided by the magnitude of the particle's momentum. This hypothesis was at the basis of quantum mechanics, and the quantity is now called the de Broglie wavelength.