Work out Aerobically versus Anaerobically
Since the term endurance refers to the ability to sustain an activity for a long period of time, to build more muscle endurance you need to do activities that can be done for extended periods. Aerobic exercise refers to exercise that is done at a level that allows the body to utilize oxygen. Oxygen breaks down glycogen which provides energy for the body to continue with the exercise. Working at a level that both challenges you but is also not so demanding that you can not do it for long will help you to increase your endurance.
In contrast, anaerobic exercise occurs when you work out so hard and fast that the body can not supply enough oxygen. This means that the body no longer has the byproducts of glycogen available for fuel. Given this you will quickly run out of energy and will no longer be able to continue with the activity for much longer. While anerobic exercise will build more muscular power you will not be able to train at this level long enough to increase endurance.
An example of aerobic exercise would be running at a moderate pace. An anaerobic exercise would be sprinting.
Use the Overload Principal
In order for the muscles to get stronger they must be challenged to handle work loads that are more demanding then they are accustomed to. This means that you must use weight that is heavy enough to really challenge the muscles. Also, as you continue to work out you must continually increase the weights as you become stronger. A good rule to use is the "10 Rep Max." This involves finding the amount of weight that allows you to complete 10 repetitions with good form before becoming too tired to continue. If you are able to easily do 10 or more repetitions then the weight is too light. If you can only do six or seven repetitions and then find yourself struggling, the weight is too heavy. Experiment until you find a weight where you can just do 10 repetitions with good from. Then stay with that weight and work up to three sets of 10. When that becomes easy, increase the weight.
Do Circuit or Interval Training
Circuit or interval training is a great way to challenge the body and build muscle endurance. This type of training rotates between fast aerobic movements and slower weight training movements. For example, you might start with two or three minutes of jumping rope or other aerobic activity. Then you would immediately follow that with a weight lifting exercise such as a bench press. You then go back to two or three minutes of aerobic exercise and then another strength training exercise. Repeat this sequence for your entire workout.
Vary Your Routine
If you continually do the exact same workout the exact same way day after day you will eventually reach a point where you no longer gain benefits from your program. This is because the body has an innate ability to adapt to your routine and since it knows what's coming, the body is no longer being sufficiently challenged. To get the most from your workout you must do a variety of exercises that challenge you to move in different ways.
Another way to cross train is to do the same exercise but with a different type of resistance. For example if you always work the chest by doing a bench press, try using a chest press machine or doing pushups on some days.
One sure fire way to watch your endurance fail is to overtrain. More is not always better. If you work out too hard without enough rest in between sessions you will eventually burn out. After lifting weights your muscles need 48 hours to recover. When you lift you create tiny tears in the muscle fibers. Rest allows the fibers to repair themselves even stronger then they were before your session. This is a healthy and productive process to help improve muscle endurance. However if you lift two days in a row, those muscle tears will not have time to heal and you will actually become weaker.
The same is true for aerobic exercise like running. Going at it too much or too hard can have detrimental effects. It is well known that if aerobic exercise is too intense you can actually decrease the efficiency of the immune system. Training too hard will cause you to become run down and weaker.
One sign of overtraining is exhaustion. If after your workout you just need to sleep or if the day after your session you can hardly move, you are probably overtraining. The right intensity of exercise should give you energy, not deplete you of it.