Hi Justin,
Endurance in a nutshell is your body's ability to handle increasing levels of stress over longer periods of time. This will consist of improving both your aerobic and anaerobic capacity.
Increasing your aerobic capacity generally consists of increasing the intensity of your workouts in length and duration over time. In short, you gradually build up your speed work and distance. This will develop your metabolic processes that are oxygen dependent, and that require you to breath during your runs. Generally you are using glycogen and fats as fuel, and it is important to have these in your diet, and perhaps more important to replace some of them after the workout in the form of carb rich electrolyte drinks like gatorade
Anaerobic metabolism is not oxygen depenedent (and you may notice your breathing rate decrease dramatically)
but can only be maintained for short periods of time, like your kick at the end of a race. You can develop this through higher intensity intervals/sprints and weight work outs. A bi-product of anaerobic metabolism is lactic acid, which can build up in your muscles and lead to soreness.
If that sounds like a lot of technical talk, the reality is you need to keep pushing the envelope. A little farther, a little faster, good diet and some knowledge about what is going on with your body. In time you should see progress if you keep striving to do better.
Hope that helps