How fast your heart beats during exercise is merely a reaction to how hard you are working. The harder you work, the more oxygen your muscles ask for to keep working. Oxygen is carried by blood, blood is moved around the body by the heart, so the more your muscles work, the harder the heart works to try and keep up with supplying the demand.
When you train at a level where the heart runs fast but keeps up with demand, you are training 'aerobicaly': literaly 'with oxygen'.Because supply meets demand, this means that you can train for prolonged periods of time (jogging, cycling, swimming etc)
When you train at a level of effort that means the heart cannot keep up with oxygen demand, you are training 'anaerobicaly': 'without oxygen'. This causes you to 'gas' or 'burn out' after a short burst of effort (sprinting, heavy lifting, HIIT etc)
Heart rate can be monitored in both types of training- for aerobic it allows you to guage and monitor your effort through the whole workout, and for anaerobic training, you measure how quickly you recover and repay the 'oxygen debt'
Heart rate monitors are not cheap, but can be usefull, but for many, going by percieved level of exertion is just as effective.