What is the difference between longitudinal and transvere waves?

 Longitudinal Waves-- 

In a longitudinal wave, the motion of the medium is parallel to the direction of the wave . This means that the particles move left and right which in turn makes the other particles start to oscillate. Sound waves are longitudinal waves. Another example of a longitudinal wave is a P wave or primary wave during an earthquake. P waves are also known as compressional waves, because of the pushing and pulling they do. See information on P waves

Transverse Waves

A transverse wave is a wave in which the motion of the medium is a right angles to the direction of the wave. 

A wave on a rope is a transverse wave. Light and other electromagnetic waves are also transverse waves.

Another type of transverse wave is the S wave or secondary wave, which is the second wave you feel in an earthquake. See information on S waves

Combination of Waves will produce Surface Waves 

Some waves cannot be described only as longitudinal or transverse. They are instead a combination of the two waves. An example of such a wave is a surface wave. Surface waves, as the name implies, occur at the surface of two material. 

Below are videos of surface waves caused caused during an earthquake.

Rayleigh Waves

A Rayleigh wave rolls along the ground just like a wave rolls across a lake or an ocean. Because it rolls, it moves the ground up and down, and side-to-side in the same direction that the wave is moving. Most of the shaking felt from an earthquake is due to the Rayleigh wave, which can be much larger than the other waves. 

Love Waves

The Love wave is the first kind of surface wave during an earthquake. The Love wave was named after A.E.H. Love, a British mathematician. It's the fastest surface wave and moves the ground from side-to-side.

Sources and Readings: 

Longitudinal and Transverse Wave Motion -- From Kettering University

UPSeis --An Educational site for budding seismologists

Transverse and Longitudinal Wave Applet --compares what the particles in the medium are doing to the direction the wave is traveling.