While, Dylan's Candy Bar is enormous and fancy (it is behind Bloomingdale's, after all), and the candy section of FAO Schwartz is famous, Economy Candy (Rivington between Ludlow and Essex) is wonderfully old-school, cramped, and fun. Since 1937--when they opened as one of the era's typical corner candy stores--they've been selling inexpensive candy, nuts, and dried fruit from all over the world.
I love checking out the candies from my childhood: Mary Janes, candy buttons (which I didn't really like, but was generally allowed to have for whatever reason), Atomic Fireballs, Red Hots, Dubble Bubble, wax bottles and lips, and--thankfully--candy cigarettes. I linger over the reasonably-priced imported British candy bars (I loved buying Cadbury bars from the machines in the London tube back in my college days), and usually pick up some Leone Polar Strong mints (a.k.a. to my friends as my "rabbit turd" mints), to which I'm totally addicted.
It's worth popping in if you're on the Lower East Side. Like many of the area's other holdouts (Katz's Delicatessen, Yonah Schimmel's Knishery, Russ & Daughters), it's an important window to our past, and I hope they stick around to sell sweets for another 70+ years.
Economy Candy, 108 Rivington Street between Ludlow and Essex.