April - October, December
Easter in Vienna kicks off the tourism season there that lasts through the summer and into the cool, breezy months of September and October, peaking in July and August. One thing you won't get in July in August is a lot of locals; Austrians tend to vacation during those two months. Still, you'll see the city with all the imperial gardens in bloom—and trust me, you want to see that. December is also a popular time, as Vienna comes to life with Christmas markets and Christmas-themed entertainment.
It's hard to declare a definitive off-season for Vienna, as it's really an all-year city, but these are two months when you might get a deal on a hotel.
January - February
People who plan a Vienna vacation for the Christmas festivities often stick around for Ball Season, which kicks off December 31st and lasts through February. Balls are not exactly tourism-focused events, but I'd certainly recommend you dance at one someday. This is a fun time to visit and watch people in tuxes and floor length gowns fill the cafes late at night and show up dressed up to the nines even at sausage stands.