Speaking non-technically, a time-domain graph shows how a signal changes over time, whereas a frequency-domain graph shows how much of the signal lies within each given frequency band over a range of frequencies. A frequency-domain representation can also include information on the phase shift that must be applied to each sinusoid in order to be able to recombine the frequency components to recover the original time signal.
The frequency domain relates to the Fourier transform or Fourier series by decomposing a function into an infinite or finite number of frequencies. This is based on the concept of Fourier series that any waveform can be expressed as a sum of sinusoids (sometimes infinitely many.)
A spectrum analyzer is the tool commonly used to visualize real-world signals in the frequency domain.
In using the Laplace, Z-, or Fourier transforms, the frequency spectrum is complex, describing the magnitude and phase of a signal, or of the response of a system, as a function of frequency. In many applications, phase information is not important. By discarding the phase information it is possible to simplify the information in a frequency domain representation to generate a frequency spectrum or spectral density. A spectrum analyzer is a device that displays the spectrum.
The power spectral density is a frequency-domain description that can be applied to a large class of signals that are neither periodic nor square-integrable; to have a power spectral density a signal needs only to be the output of a wide-sense stationary random process.