To attempt to behave as a gentleman would. A gentleman, by definition is polite, courteous and drinks beverages with his "little" or "pinky" finger sticking out. Of late the term "Like a gentleman" has tended to be tack on to the end of a sentence for humourous effect, especially in the context of drinking alcoholic beverages. See also like a lord.
I'm going to drink this beer like a gentleman.
I was drunk last night, drunk as a gentlemen.