Should I take breaks when exercising?

Take frequent breaks. Even tennis pros rest between sets. Taking a rest doesn't mean that you have to completely stop all activity (although it may be advisable sometimes). Just rest the body parts that are working hard and are susceptible to injury.

Yes you should take breaks when exercising. For various reasons such as cardiac, muscular endurance, injury/pain history and other things you should consult with a qualified health practitioner to determine when to take the proper break according to your individual health status.

Whether or not to take breaks while exercising is dependent on your current health and fitness levels, the intensity of the exercise and your goals.  If you start to become out of breathe, experience dizziness, or extreme fatigue then breaks are appropriate. If performing resistance training exercises or exercises of a high intensity in nature requires breaks in between sets is important to prevent injury. The more intense the activity the longer and more frequent the rest periods should be. 

Absolutely! It is important to give your muscles a chance to rest and allow them the repair and get stronger. Without proper recovery time, areas of wear and tear become weak links and are more prone to injury and will less likely be able to show strength gains.