Because of its desert location, the best times to visit the park are in spring (late March through mid-May), and fall (mid-September through November).
Springtime in the desert is cause for a mad dash into fertility - desert plants, normally drab and reserved, blossom into a riot of colors. Starting late in March, and continuing through June, the desert bloom adds a new dimension to the normally austere desert color scheme; May seems to be the peak of most of the flower activity. Spring is sometimes cloudy, though not usually wet.
Summer is hot and sunny, punctuated by sudden, severe thunderstorms. This is also the most popular time of the year for tourist visits to the park. Avoid the crowd and the burning sun if you can - the park is much better viewed when you're not trying to stay cool. And avoid the trails during storms - you're prone to get struck by lightning standing on top of the fins, or washed out standing in the canyons.
Fall brings moderation of the temperatures, and also drier weather (okay, it's a desert - it's usually dry - but fall is more reliable for blue skies than any other season). Unfortunately, most of the plants are well and truly dried out by this time of the year, and the vegetation looks as bleak as it does through winter. Fall is very popular with the more local crowd.
Winters can get chilly in the desert, though afternoon temperatures are often reasonable. On a recent trip in February, the temperatures hovered in the 40s and 50s during the day, but the overnight temperature drop was significant, and early morning hikes were less than warm (I'll omit a description of standing out on the desert plain waiting for the sunrise with the morning winds blowing...). There is sure to be snow or ice in the more shaded areas of the park, though not much.
Something to consider when timing your trip: the Fiery Furnace area of the park (which requires a guide or a permit), is only manned from April through October. If you want to see this section of the park, Winter and the months surrounding it are not available to you.
Also to consider: Moab is home to several mountain biking and offroad festivals. Unless you like fighting overpopulation, try to avoid these very popular events. I have been caught in mountain biking events in mid-October (Canyonlands Fat Tire Festival) and in mid-May (Behind The Rocks Festival)