What criteria are used for classifying composers?
Primarily the time period in which they composed but a composer can span periods and the boundaries can be blurred anyway.
It's easy to say JS Bach is baroque and Mozart is classical, but what about Bach's sons? Is Beethoven entirely classical or early romantic?
Other criteria that could be used would be what forms they composed in and what harmonic vocabulary they used. Even instrumentation and orchestration can be used as a criterion.
Some of Mozart's chromaticism seems before its time in the classical era. Much of Beethoven's later work clearly belongs in the romantic era. You could argue Wagner kicked off a new era just with a single chord progression. Stravinsky's works span three distinct styles.
(its been a long time since I looked at any of this! Sorry if its not clear)
The most useful criteria are, unfortunately, circular. - Years active - Contemporaries
For example, Bach is clearly baroque. I've heard it said that his birth and death dates define the Baroque period.
Sometimes it is difficult to classify a composer as a whole, and the style of the piece needs to be taken into account. Beethoven for example starts in the Classical period, and ends in the Romantic.
A composer, throughout his or her life, will compose whatever music suits him or her. At some time later in history, a musicologist or historian will stereotype a composer as belonging to a certain style or school of music. The one constant is that composers, while alive, resist and resent being "pigeon-holed" or labeled as being a certain "kind" of composer.
When I was in music school in the 1980s, I listened to music by living composers who were popularly referred to as being of the "minimalist" style. The first thing I found out is that all of the composers in question hated being called minimalist composers. You could say that the definition of a minimalist composer in the 1980s was somebody who hated being called one.
From my own listening experience (I'm referring to classical music in the European tradition here) I find that geography is also very important. You'll perhaps be able to guess whether a piece is from say the French or German tradition, or from Russia or Italy, even if composers were active during the same broad period, just by listening to it.
BTW, one nice thing about listening to radio stations (and podcasts) on the internet is that you can suite your own taste, because these geographic differences still give partly different flavors to the different national radio stations (think of Europe), so you can learn and choose accordingly.