What is a wave that appears to stay in one place?

A Standing Wave, the principle of superposition states that : The resultant of two or more superposed harmonic vibrations is simply the sum of the displacements of the individual vibrations.

To understand better what is a stationary wave, you should understand how stationary waves are formed.

Check out Melde's set up.

Melde, set up an apparatus, where one end produced a wave when the oscillator was switched on, the wave then hit the pulley and bounced back. This wave hit the incoming new wave from the oscillator and since they had the same characteristics (same wavelength, speed, frequency) and were in the opposite direction they created a stationary wave.