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Waves can occur in several different forms: in the ripples of a pond when a stone strikes the surface, in the vibrations of a string, in the sound we hear, and in the light we perceive. Waves may propagate through a medium (mechanical) or may be self-propagating, as is the case in electromagnetic waves.
You can boil all of these cases down into two types of waves: longitudinal and transverse. The former is a wave where the energy propagating from some disturbance causes oscillations to occur perpendicular to the direction that the wave travels.
Consider shaking the end of a long rope in quick up and down motions. You would notice small waves traveling in succession down the length of the rope. Now consider taking a Sharpie (one with a fancy color like chartreuse or taupe because why not live life to the fullest?), marking a single point anywhere on the rope, and making those waves in the rope again. If you pay attention to the dot, you will notice it simply oscillates up and down over the same point, on a vertical axis. This demonstrates exactly the mechanism of a longitudinal wave, where the wave appears to travel down the rope while really every point on the rope stays in its proper place on the rope and simply oscillates vertically.
The transverse wave is defined by some disturbance causing rarefactions (points of expansion) and compressions of a medium as energy propagates through. Any point or particle in that medium oscillates around its natural, resting position in parallel with the direction the wave travels.
To illustrate this idea, consider holding the two ends of a slinky horizontally in your two hands. If you were to lightly tap then pull one side of the slinky, you would notice a little pulse, with a patch of more densely packed slinky rings followed by stretched out slinky rings, travel from one hand to the other. Now if you were to pick any ring on the slinky, you'd notice it sort of speed up as the pulse passed over it, then pull back and move in the opposite direction before returning to its starting point. It oscillates around its natural position along the same axis as the direction of the wave, which is characteristic of transverse waves.
The two types of waves differ in the way the energy from the disturbance affects any particle in the medium the wave travels through– in a longitudinal wave the particles oscillate perpendicular to the direction of the wave and in a transverse wave the particles oscillate parallel to the direction of the wave. For non-mechanical waves such as electromagnetic waves, the same principles apply: longitudinal is characterized by oscillations perpendicular to the wave's direction and transverse is characterized by oscillations parallel to the wave's direction.