The trumpet has evolved very slowly over the years. Early natural trumpets had no valves, and could really only play melodies in the utmost of registers. They were often used for ceremonial purposes to play fanfares of sorts. Over the years keys were added to the trumpet in order to create notes that were missing in the harmonic series. Several composers of the time, such as Haydn and Hummel, took advantage of these keyed trumpets. The most important advance for the trumpet was the invention of the valve. The valves allowed for a brass instrument to be capable of playing a full scale over the range of the instrument, and hence led to today's modern trumpet.

Here is a more in depth history of the trumpet take from the New Harvard dictionary of music: http://www.petrouska.com/historyofthetrumpet.htm

The cornet is an instrument that is very similar to today's modern trumpet. One of the main differences between the two is that the bore, or inner diameter of the instrument tubing, has more of a conical shape in the cornet, while the trumpet's is more cylindrical. The conical bore leads to a more mellow sound. The cornet first appeared in France around 1828 when valves were first being added to brass instruments. Around the turn of the century cornet came to forefront in solo playing, and many bands had in them a cornet soloist. Some of the great cornet players of the time were most notably Herbert Clark, Bohumir Kryl, Allesandro Liberati, Walter Rodgers, and Jules Levy. While the cornet is still around today, it has mostly been replaced by the trumpet. If you are interested in hearing some good cornet playing I'd recommend the two CDs:

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