A standing wave (physicists, help me out here) is a pattern of distortion in a substance or object that is the product of two or more interfering waves. To the viewer, the pattern appears immobile if the interfering waves are in the proper synchronization; this is a 'standing wave' because the wave pattern produced won't move until the frequency of one or more of the interfering waves is changed.

You can create a standing wave with a rope by keeping one end fixed and moving the other end up and down. You create a wave moving away from you and it reflects back from the fixed end, creating two waves of equal magnitude and direction but opposite directions. Each end of the rope is a node. If you double the frequency that you move the rope you will create a new node in the middle of the rope.

Standing waves come up in lot of situations. The vibrations of a drum skin for example are a very common example of standing waves. This example of waves on a rope is something you will find in almost all text books and it helps in understanding but its actually very difficult to observe. Try it!!
Plucking a guitar string, or banging one of those really flexible skins is simpler to do.