An EEG (electroencephalogram) "is a recording of the electrical activity of the brain from the scalp" (McGill). The recorded wave patterns reflect the electrical activity taking place in the subject's brain and fall into one of four classifications. They are Alpha, Beta, Delta, and Theta states.
Each of these patterns has its own corresponding amplitude and frequency pattern. Delta waves, for example, tend to have the slowest frequency and the highest amplitude. On the opposite end of the spectrum, Beta waves have the fastest frequency and the typically smallest amplitude (the waves come smaller and faster as the needle monitoring them moves short distances back and forth very rapidly).
To understand this correlation, we can observe when subjects would experience such patterns of brain activity.
Delta waves are observed in adults in a state of deep, non-REM sleep. At this time, "neurons, which are not engaged in the processing of information, are firing all at the same time." This synchronicity results in waves that are "large and slow."
Beta waves are observed in adults that are awake, alert, and in a state of rapid thought. They can be accentuated by drugs such as barbiturates. Neurons are firing and communicating with each other rapidly and in an asynchronized manner. Because of this, the amplitude of the brain waves is smaller, but much more rapid.
Just think of how different activities - deep sleep versus solving a difficult problem - require different output from your brain cells, much like different physical activity requires different output from your heart; brain waves will either be slow, regular and strong (like your heartbeat would be while sitting down), or quick and pulsing (like your heartbeat would be while sprinting).