Brain waves are a colloquial term for EEG recordings. EEG recordings are gross potential recordings, in other words, they represent the responses of thousands of neurons together. Much of the background is stochastically determined, meaning it represents, basically, random activity. In other words, oftentimes neurons fire stochastically and they cancel out each others contribution to the EEG, resulting in random ('white') noise.
However, certain parts of the brain do exhibit synchronized activity in certain frequency bands (Fig. 1). Oftentimes a time-to-frequency analysis is necessary to visualize these activities, e.g. by FFT. Certain stages of sleep, most notably slow-wave deep sleep, is characterized by gross synchronized, slow-wave activities across the cortex, showing up in the raw EEG. In fact, simply closing ones eyes has marked effects on the cortical synchronization on the raw EEG. This is caused by the fact that certain swaths of the cortex become synchronized, resulting in more clear responses.
In general, to come to your question, gross potential recordings like EEG thus are plagued by background noise. Synchronized activity increases the amplitude of brain waves and hence increases their amplitude. Amplitude is hence directly related to the level of synchronization. This shows up as larger peaks on the FFT.
What synchronization exactly does to the brain boils down to down to is the classic interpretation of EEG frequency bands (Table 1).
Fig. 1. EEG waves. source: Neurosky
Table 1. EEG frequency bands and their associated mental state. source: Liang et al. (2016)
- Liang et al., J Sensors (2016), 1831742