The peak tourist season runs from June through August at the Walt Disney parks. It features the biggest crowds, the longest lines and oh yes--heat as only Southern California/Central Florida can provide. Similarly crowded periods take place during the two weeks around Christmas, the nebulous length of spring break (mid-March through the end of April) and holiday weekends like President's Day or Columbus Day. Disney responds by providing additional amenities during these periods: parades, nightly fireworks shows and longer hours to handle all of the crowds. That said, if you book a Disney vacation during this time of year, plan for additional days to see everything or be prepared to prioritize the attractions you want to visit.
Offseason (what Disney calls "Value Season") runs from January to mid-March and September through November. The kids are in school, people are at work, and the crowds are much more manageable. Conversely, the parks don't stay open quite as late, parades aren't as fresh, and fireworks displays only occur on weekends. On the plus side, you can see much more of the parks--with a lot less stress to boot--and the Anaheim/Orlando weather remains comparatively warm, even in the middle of January. Disney also tends to offer more discounts during the offseason, both on tickets to the parks and on rooms at Disney hotels. If you can manage it, try to book a vacation in the offseason rather than the peak season when everyone else is trying to visit the park as well.
As a general rule, Disney parks are more crowded on Saturdays and Sundays than they are in the middle of the week. Some estimates hold that three times as many people come on Saturdays as they do on Wednesdays or Thursdays. Booking vacation time becomes a matter of convenience--with work schedules and school schedules making things more difficult than on weekends or holidays. Remember, if you go during the offseason, the parks close early on weekdays. But the offseason does make for a smoother and less stressful visit, giving you more space to enjoy the park at your leisure.
