Ich habe diese Sätze gefunden :
Wo ist bitte das Cafe? - Das ist an dem Rosenplatz.
Wo ist der Markt? - Auf dem Kirchplatz.
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possibly relevant: german.stackexchange.com/questions/2979/…– Vogel612Jan 18, 2014 at 17:23
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1 Answer
Basically, the difference is clear:
"An dem (oder am) Rosenplatz"
means that the Café is located adjacent to the place.
"Auf dem Platz"
means on top of it.
A "Markt" (market), not to be confused with "Supermarkt" (market center) in an European city, is comparable to a bazar, so it consists of "Stände" (counters with a tent) that are placed on the market place. See the image from a market in Germany (From de.wikipedia article "Markt"): 
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Colloquially, sometimes the wrong preposition is used, but that is bad style.
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1I'd like to add that "Markt" is also used in an abstract sense as "market" ... as in stock market... you will find "am Markt" in that context a lot.– EmanuelJan 18, 2014 at 20:00