A longitudinal wave is one whose energy propagation is parallel to oscillation (vibration). Examples include a sound and p waves (seismic waves that travel relatively fast).
A longitudinal wave transfers energy in a way that involves compression and rarefaction. One can use a slinky to demonstrate that by applying a force on one end which will be transferred through the slinky (as can be seen below)
The areas where the 'loops' are bunched up is compression, the parts where they are separated is called rarefaction.
Transverse
A transverse wave is one whose energy propagation is at 90° from the direction of oscillation (vibration) - it is perpendicular.
Most waves are transverse - all the waves in the electromagnetic spectrum are, and travel at the same speed in a vacuum. Water and S waves (another type of seismic wave) are types of transverse waves.
One can easily measure the amplitude and wavelength of a transverse wave. Just remember that amplitude is measured from the point of equilibrium (dotted line in the diagram below)
You can move a rope attached to a thin tree (or something like that) and move the rope up and down, and you'll notice that the tree moves. The faster you move the rope, the faster it moves.
See http://www.bbc.co.uk/schools/gcsebitesize/science/aqa/waves/generalwavesrev2.shtml
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