The most wonderful months of the year in Chicago are May, September and early October. The differences between May and October are minimal weather-wise but the scenery differences are extraordinary. The sun comes in from a different angle in May and the light is so much brighter and the colors of the leaves are a much lighter green - they look so new. In the fall, the sun has moved and you can tell the difference immediately, the leaves are changing colors and what's left of the green ones are a much deeper green. There is usually low or no humidity in these months, the nights and mornings are cool - no need for air conditioning or heat just yet. You can take a walk and not work up a sweat, sit outside to read or talk and not have to swat mosquitos or bees, build a fire in an outdoor fireplace and actually feel a need for it. There are a lot of people who like hot summers for swimming, sunning and other outdoor activities, but the humidity in Chicago in the summer can be a real drain. There are a lot better hotel deals during these times as well - not the height of the tourist season and you can catch a Cubs game in the Spring and September and watch the Bears in September/October. Best time for viewing the fall colors in the area is in early October - they are just beginning to change and not too many leaves have fallen off yet - just enough to make it interesting. Chicago is a beautiful city and seeing it in the two different lights is a wonderful contrast.
While the shoulder seasons are lovely, if you are there during the summer (and summers do vary heat and humidity wise) do bring a bathing suit and beach towel so you can try a long walk or bike ride along the lake front (you can rent one at Navy Pier and other locations) and just jump in the lake when you get too warm. Summer brings lots of festivals: neighborhood street fairs and markets, art fairs, book fairs, music festivals in Grant Park (almost one every weekend), classical music outdoors at Grant Park and Ravinia, and of course, the Sox and the Cubs.
And for a counter point, December is a great time to visit Chicago - the weather is not bitter cold yet, and the crowds, shopping, and lights on Michigan Avenue and all around downtown are festive and bright. Other Christmas highlights are the big tree and German Christmas Markets and Daley Center and the Zoo Lights festival at Lincoln Park Zoo.
Unlike many cities in Europe and Canada, Chicago doesn't really have a late winter/Carnival attraction. But if you don't mind the cold weather, it's a good time to look for hotel bargains and enjoy the arts, which are typically in full swing at this time.