Thankfully, there is not a "one size fits all" exercise routine that must be followed. Simply put, your fitness goals dictate the exercise order. This gives you the freedom to focus on the areas of your interest and needs. For the average recreational athlete whose goal is to achieve overall fitness and good health, exercise order is a matter of personal preference and comfort.  Competitive athletes seeking improvements in performance and skill, however, need to pay closer attention to the order in which they train their muscles and cardiovascular system to ensure they are getting the most from their exercise session. Here are a few points to consider when deciding the proper protocol for you:


Improvements in Aerobic Capacity
If your goal is to increase cardiovascular endurance, then aerobic training should occur first in the exercise session. Weight training prior to the aerobic phase could fatigue your muscles. As a result, you may need to either reduce your exercise intensity or prematurely end your aerobic workout, both of which could impair your aerobic capacity in the long run.

Improvements in Muscular Strength and Power
Weight training should occur first in the exercise session when greater muscular strength and power are desired.*  Weight lifting is most effective when your muscles can be challenged at maximum ability.  Performing cardio training first will lower the threshold and limit the amount of weight that you can lift. Furthermore, your ability to maintain proper form while lifting may be compromised if your muscles have been fatigued from aerobic exercise, increasing your risk for injury. Improved muscle recovery is another advantage of weight training prior to aerobic training. The aerobic exercise will help to enhance blood flow and nutrients to the taxed muscles and will facilitate waste removal.

Other Points to Consider
Regardless of whether you do cardio or resistance training first, you should always start with a warm-up. The warm-up should consist of 5 to 10 minutes of mild aerobic activity, such as walking or cycling. A warm-up allows your muscles and cardiorespiratory system to adequately adapt to the increased demands of the training session. A cool-down should also be performed at the end of the exercise session. This will help to facilitate blood flow back to the heart and brain, reducing your risk for untoward events, such as abnormal heart beat or fainting spell.


*Note: during the resistance training phase, exercises to improve muscular power should be performed before exercises to enhance muscular strength.


Before beginning an exercise program or increasing the intensity level of a current routine, a physician's approval should be obtained, especially for older adults and those at risk for or who currently have chronic health conditions. 
 
 
References
American College of Sports Medicine Position Stand: Progression Models in Resistance Training for Healthy Adults
 
American Council on Exercise