longitudinal wave
Push and pull one end of a stretched slinky along its length and a squeeze, a bunched-up region of coils, travels down the toy, followed by a stretched-out region. A longitudinal wave is one where the wiggling is back and forth along the same line the wave travels. The everyday case is sound. The same question comes up: which way do the particles move relative to the wave? Here, along it.
In a longitudinal wave the particle displacement is parallel to the direction of travel. The medium is alternately squeezed into a compression (particles bunch together and the pressure rises) and stretched into a rarefaction (particles spread apart and the pressure drops), and these compressions and rarefactions march forward at the wave speed. There are no crests and troughs in the sideways sense, instead there are regions of high and low density and pressure.
Sound in air is the classic longitudinal wave: your vocal cords or a loudspeaker push on the air, sending compressions to your ear. Longitudinal waves can travel through solids, liquids, and gases, because any material resists being compressed. An honest note: some real waves are mixed, surface water waves and seismic waves carry both transverse and longitudinal parts, so a purely longitudinal wave is an idealization for cases like sound in a gas.
A drum: the skin pushes the nearby air into a compression, then pulls back to make a rarefaction. This pattern of squeezes and stretches races to your ear at about 343 m/s.
In a longitudinal wave, particles oscillate along the direction the wave travels.
Sound in air is longitudinal, a pressure wave rather than an up-and-down one. So drawing sound as a sideways squiggle is a graph of pressure, not a picture of the air moving sideways.