You may strongly prefer to go in a certain season but the size of the travel industry makes it flexible enough that you will find a trip that will suit you regardless of when you want to travel. Even so it may still be easier to go in one of the three seasons that will best suit your requirements so here I will discuss the pros and cons of each season in turn.
Peak season; June-August
Pick this one if you love to be around others in the thick of the action and don't mind mad rushes and busy airports. This is also the hottest season so be prepared with the high strength sun block! Another advantage of this period is that all the tourist attractions will base their events around this time and so there will be most going on for you to enjoy during this time of year. If you want to make the most of your holiday and cram in as much as you can this may be the season for you. Bear in mind this will be the most expensive time of year to travel but if money is not an issue go for it!
Shoulder season; April-June and September-October
This season can be thought of as getting the best of both worlds. The weather is fair but not overbearingly hot and there is enough going on to keep you entertained without being frenzied and overwhelming. Exactly when shoulder season applies will depend on your destination as different parts of the world will have different weather cycles of which the on and off seasons are based. Check online or with your local travel agent to find out the dates for the specific place you wish to travel.
Off Season; November-March
The benefits of the off season could quickly be summed up as the opposite of the peak season. You will save money finding a budget holiday and the local shops and attractions will be glad of your custom during their slower period. At the same time note that there will be a lot less going on though this may be an advantage for you if you prefer lazy days on the beach with nothing to do. The weather will be a lot more mild or even cold depending on where you go though if you are sensitive to heat or burn very easily again this could be what you are looking for. Be aware the hours will be shorter during this periods so plan your activities carefully.
Hopefully this gives you a better idea of when to travel and the benefits and drawbacks of each season. Good luck planning your next adventure!