A sound spectrum displays the different frequencies present in a sound.
Most sounds are made up of a complicated mixture of vibrations. (There is an introduction to sound and vibrations in the document "How woodwind instruments work".) If you are reading this on the web, you can probably hear the sound of the fan in your computer, perhaps the sound of the wind outside, the rumble of traffic - or perhaps you have some music playing in the background, in which case there is a mixture of high notes and low notes, and some sounds (such as drum beats and cymbal crashes) which have no clear pitch.
A sound spectrum is a representation of a sound – usually a short sample of a sound – in terms of the amount of vibration at each individual frequency. It is usually presented as a graph of either power or pressure as a function of frequency. The power or pressure is usually measured in decibels and the frequency is measured in vibrations per second (or hertz, abbreviation Hz) or thousands of vibrations per second (kilohertz, abbreviation kHz). You can think of the sound spectrum as a sound recipe: take this amount of that frequency, add this amount of that frequency etc until you have put together the whole, complicated sound.
Today, sound spectra (the plural of spectrum is spectra) are usually measured using
a microphone which measures the sound pressure over a certain time interval,
an analogue-digital converter which converts this to a series of numbers (representing the microphone voltage) as a function of time, and
a computer which performs a calculation upon these numbers.
Your computer probably has the hardware to do this already (a sound card). Many software packages for sound analysis or sound editing have the software that can take a short sample of a sound recording, perform the calculation to obtain a spectrum (a digital fourier transform or DFT) and display it in 'real time' (i.e. after a brief delay). If how have these, you can learn a lot about spectra by singing sustained notes (or playing notes on a musical instrument) into the microphone and looking at their spectra. If you change the loudness, the size (or amplitude) of the spectral components gets bigger. If you change the pitch, the frequency of all of the components increases. If you change a sound without changing its loudness or its pitch then you are, by definition, changing its timbre. (Timbre has a negative definition - it is the sum of all the qualities that are different in two different sounds which have the same pitch and the same loudness.) One of the things that determines the timbre is the relative size of the different spectral components. If you sing "ah" and "ee" at the same pitch and loudness, you will notice that there is a big difference between the spectra.