When many people think of music from the past, a common misstatement is that it all is "Classical Music." There is nothing wrong with this, as it is a convenient way to refer to the numerous periods of music from the past. I really don't think it is necessary to know the exact period of each work of music we listen to or play. However, I feel that every musician should have a brief knowledge of the layout of the periods of classical music.
The major periods of music in the order they occurred are: the Baroque Period, the Classical Period, the Romantic Period, and the Contemporary (The 20th Century) Period.
The Classical Period, as its name suggests, showed a shift in the focus of music to emphasize the works and culture of ancient Greece. This shift is believed to have started around 1750, and the style continued into the early 1820s. Many developments were made during this period, and many famous composers emerged from the woodwork.
In the mid-1700s, the mainstream musical audience shifted. Instead of primarily writing for the church, many composers began to accept commissions from monarchs and diplomats. The patrons would pay the composers to write and present "chamber music" for them, and preferred light, elegant music over the complex melodies of the Baroque Period.
It was during this period that Vienna became the musical center of Europe. Many composers trained under Viennese musicians and composers, and a common style began to emerge across Europe. The "old rules" of the Baroque Period were modified and followed loosely. The composers began to focus on the art of music rather than the form.
Early in the period, Johann Stamitz and his fellow composers in Mannheim, Germany, began to work on the art of orchestration. They based their forms on Baroque forms, but made a few major changes. Chords were played in unison and figured-bass was written out for instruments instead of being left to performer interpretation. It was through these changes that the symphony was developed.
This period also saw the first transcriptions of German Lieder(songs) as an art form. Franz Schubert wrote many compositions based on the folk songs of Germany, many of which are still standard vocal solo literature today. Wolfgang Amadeus Mozart and Franz Joseph Haydn made further developments to the symphony, along with Ludwig van Beethoven.
Piano sonatas also became a staple of the era. Two of J. S. Bach's sons, Carl Philipp Emanuel and Johann Christian, focused on works for the keyboard. Their numerous concertos and sonatas influenced other great keyboard composers of the age, and their influence is most apparent in the piano sonatas of Beethoven.

Some classical era composers (left to right):
Haydn, Mozart, Beethoven and Schubert
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It is beleieved that the climax of the Classical Period was at the height of the composing careers of Beethoven, Mozart, and Haydn. Their symphonies are some of the most commonly performed and heard, even today. Their works show the influences of the period by focusing on being musical instead of heavily ornamented scores.
As the period came to an end at the start of the nineteenth century, a brief whisper of Romanticism began to float on the wind. Franz Schubert began implementing a few styles that differed from the rest of the composers of the time, leading, perhaps, to the shift to the Romantic Period.
It is interesting to note, however, that this period that called for a focus on elegant music instead of elaborately decorated notation is the namesake for much of the music that preceeds the twentieth century. Much of what occurred in this period was carried through even to the "classical" composers of today.