For a wide-sense stationary random process, average power can be distributed across frequency even though the individual realization is not periodic.

The power spectral density, or PSD, is the Fourier transform of the autocorrelation function:

$$S_X(f)=\int_{-\infty}^{\infty}R_X(\tau)e^{-i2\pi f\tau},d\tau.$$

This relation is the Wiener-Khinchin theorem.

Integrating the PSD over a frequency band gives the contribution of that band to the process power. In particular,

$$R_X(0)=\int_{-\infty}^{\infty}S_X(f),df.$$

A narrow spectral peak means much of the random signal's power is concentrated near a particular frequency. A broad PSD means the fluctuations occupy a wider range of frequencies.

PSD is not the Fourier transform of one arbitrary noise waveform. It is a second-order statistical description obtained from the ensemble autocorrelation, or estimated from sufficiently representative data under additional assumptions.