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.
I can't give you a customized answer to your question about you without having any data about you. In general, if one wants to improve aerobic endurance, one needs to train at paces and durations that stimulate your aerobic energy system and delay paces and durations that stimulate your anaerobic lactic energy system. Anaeobic alactic stimuli like 30 meter sprints at 100% effort with full recovery are OK anytime.
Make sure you understand what endurance means as it relates to running. Basically if you can ran say - 3 miles at a certain pace, lets say in 24 minutes which equals to 8 minutes per mile pace. Your questions is - How can I improve my endurance? Using the 3 miles example lets say that you want to bring that time down to 7.5 minutes per mile then you have to do speed workouts and hill workouts. Simple 7 days workout, assuming that you are running at least 3 miles a day - 3 days do LSD run on flat ground tempo pace - 2 days do LSD on moderate hills easy - 1 day do hill repeats on a 200yard/meter hill tempo pace - 1 day do speed workouts do repetitions of 1/4 milex6 at race pace timed. Alternate the speed and hill workouts in between the 3 days LSD flat ground. Tempo pace means not slow but not fast either. Race pace means you run as if you are racing. Endurance = Anaerobic capacity - basically means your body's ability to do a physical activity with little oxygen. Aerobic = doing physical activity using oxygen as fuel.