Bright, warm, dark, rich are all subjective terms which describe tone "color."  Timber (pronounced TAM bur) is one of the primary acoustic characteristics that allow us to distinguish which type of musical instrument we are listening to and to notice differences between two instruments of the same type. 

When a note is played on a violin, let's say the open A string, the fundamental frequency is about 440 Hz. The fundamental frequency is the lowest frequency that is generated.  If only that single frequency was generated, it would not sound pleasant at all. It would sound pretty much like the dial tone on a telephone.  However, when a note is played on a violin, many frequencies are generated.  The most important are the harmonic frequencies which are multiples of the fundamental frequency. For the open A that's 440, 440 X 2, 440 X 3 and so on.  In addition, other frequencies are also generated that are partial multiples for the fundamental frequency.  This creates a sort of frequency soup that gives the violin it's characteristic sound or "color".  

The fundamental frequency isn't necessarily the loudest or "dominant" frequency.  The harmonic frequencies can be as loud or louder than the fundamental frequency.  For example, on the transverse flute the fundamental x 2 frequency is the dominant frequency.  In cases where the fundamental and lowest harmonics are most pronounced, the sound is "dark." Where the low-mid tones are most noticeable the sound is "warm." Where the upper frequencies are more noticeable the sound is "bright." Where the mixture of harmonics overshadows the partial harmonics the sound is more "pure".  Where all of the harmonics and partials are more or less equal, the sound is more "muddy." 

Keep in mind, however, that tonal color is completely subjective, so there are no hard and fast rules.  The concepts are more felt by the listener, than something that can be defined on graphs and charts.