There are two different types of wave: longitudinal and transverse. When you hear the word "wave" you are probably thinking of a transverse wave.
Imagine you are floating in the sea. The waves of water are coming towards you and you bob up and down over the waves as they pass you. This type of wave is a transverse wave, the movement of any point in the wave (you in this case) is at right angles to the direction the wave is moving (towards the beach). When you are at the top of the wave you are at a peak and when you are at the bottom of the wave you are at a trough. Another example of this type of wave is a light wave (although the waves are a lot smaller).
A longitudinal wave is a one where the movement of any point in the wave is in the same direction the wave is moving. These are harder to visualise. If you push and pull a slinky in a straight line you will see areas where the slinky is more bunched up and more spread apart. These are called compressions and rarefactions. An example of this type of wave is a sound wave.