Correction: a cent is a hundredth of a semitone.
As David says, a spectrogram is a time/frequency representation with (in this case) time on the x axis, frequency on the y axis, and the strength of a particular frequency component expressed as colour or intensity.
It's computed as a series of short-time Fourier transforms, each one covering a single column (one horizontal unit on the time scale), which calculate the frequencies at which sinusoidal component waves would have to be present in order to sum to the original signal. The vertical (frequency) resolution depends on the frame size for the Fourier transform (a longer frame gives better frequency resolution), and the horizontal resolution depends on the frame size and amount by which analysis frames overlap (a longer frame gives worse time resolution).
So, you can read directly from the spectrogram (with some very substantial caveats to do with resolution limitations) the frequency of a component of the signal. If your music consisted of simple sine tones, there would be (roughly, with the same caveats) a single horizontal line for each note whose height told you directly what the frequency of the note was, and therefore what musical pitch it was at (there is a relationship between musical-note and frequency such that ascending by one octave results in a doubling of the frequency).
Musical tones have more than one "partial" frequency -- they consist of more than one frequency component -- usually at closely related frequency intervals such as integer multiples of the lowest, "fundamental" frequency of the note. These will show up in your spectrogram as a stack of horizontal lines, waving about in correspondence with any pitch variation or vibrato, with the lowest line usually giving the fundamental frequency and usually corresponding to the note's perceived pitch, and the structure of higher partials having some relationship to the timbral quality of the instrument. Because these structures can be quite complex, it's not always straightforward (indeed not always possible) to read off the actual performed pitch, especially in polyphonic music.
Percussive sounds are often also visible in the spectrogram, usually as fuzzy columns -- they generally consist of noise that is dispersed across broad frequency ranges without a regular structure.