The harmonic series is the set of notes that can be played on a natural trumpet, or a trumpet without any valves to alter length. In the harmonic series, the notes grow closer to each other as they become higher. For this reason, instrument makers before the invention of valves made trumpets of a longer length so that more notes in the series could be used. The harmonic series for a modern, shorter, Bb trumpet starts out like so:
(imagine a symbol for the treble clef down there)
         -C-          
          Bb
         ---
          G
----------------------|
          E           |
----------------------|
          C           |
----------------------|
                      |
----------G-----------|
                      |
----------------------|

         -C-
One could continue on in this series, but, for the sake of simplicity, it shall be left like so. The harmonic series is also responsible for the overtone series that make the trumpet's tonal quality, or timbre.