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1.
The scientific term for Aspirin. In many countries, notably Canada and Germany, "Aspirin" is a patented trademark of the local Bayer subsidiaries. Any non-Bayer products are labelled with the abbreviation ASA (Canadian English), AAS (Canadian French), and ASS (German). I am not making the last one up!
The term Acetylsalicylic Acid, in its entirety, is rarely used in layman's conversations.

Just as many people refer to any tissue as "Kleenex," Canadians often refer to non-Bayer ASA as "generic Aspirin" or, simply, "Aspirin."
by DFJD May 09, 2006
 
2.
The active ingredient in asprin and one of the ingredients in Alka-Seltzer. Acetylsalicylic acid (or its various translations) may be used as a generic term in countries where Bayer's trademark claim to asprin was not invalidated after World War I (e.g., Germany, Canada, Mexico).

The term may also see use where it is desired to refer to the pure chemical, as opposed to any medical formulations. It may also see use where the person saying it wants to sound scientific or very intelligent.
I took some acetylsalicylic acid for my headache.
by Oglesby Trelease July 15, 2010