Power: Power, from a technical perspective, is the rate at which work is performed. When this definition is applied to strength and condition, we can think of power as the rate at which force is being generated. Thus, a 500 lbs squat might last for a few seconds and although a significant amount of force is generated, it is generated very slowly, which taps into more maximum strength. We can compare absolute strength in the squat with true power in faster movements such as the vertical jump or the power clean, where the absolute amount of force is important, but not as important as the rate at which that force is produced. Thus, power is an emergent property of strength, speed, and (for more complex body movements) coordination.
Muscle Endurance (Short/Med/Long Durations): Muscle endurance, as mentioned in the previous post, refers to a muscles ability to maintain a certain level of work over time. At different durations of activity the limiting factor in endurance changes from the more neurological to the more metabolic. For short durations (ME-S), neural drive and the phosphagen system are limiting factors in the muscles performance.
For medium durations, the limiting factor is more metabolic, but rather than being an issue of using up energy substrates the issue is lactic acid build up and pain. During moderate intensity exercise the level of lactic acid produced by glycolysis is low enough for it to be absorbed quickly, but with high-intensity exercise lactic acid is produced faster than the body can absorb it. This lactate threshold is marked by an increased blood concentration of lactic acid, an increase in hydrogen ions, and an increased acidity which causes fatigue (perceived pain) and reduces the power of muscle contractions.
In long duration exercise, the limiting factor is the energy substrate. Thus, when intrafusal glycogen stores are depleted, there is little fuel left in the athletes tank. Endurance runners, cyclists, and triathletes are probably familiar with this unfortunate state, and they call it "bonking". Interestingly, you can bonk for a number of different reasons and feel the effects in very different ways. If you have depleted your muscle glycogen stores, your brain will feel fine, but your legs stop working. On the other hand, if you have depleted your blood glucose stores, you brain is running out of fuel and will start to slow down even when your legs are feeling fine.
Maximum Speed: Maximum speed is the top a speed a person can achieve regardless of time. It is simply the peak speed that you can hit, it does not matter if you hit that speed after o.3 s or 30s and it does not matter if you maintain that speed for 10 s or a minute. Maximum speed is an emergent property of a persons strength, speed (in the sense of neural drive), and coordination. When seeking to improve your speed any one of these abilities can be developed separately or in tandem. For instance, a sprinter might want to work on their leg strength, because more forceful contractions in the muscle will create more push off the ground and increase the distance covered in each stride. Or, the sprinter might want to work on developing the frequency and duration with which they can maintain neural drive to the muscle. Finally, a sprinter with good strength and neural drive might need to improve the efficiency of their stride by working on coordination. Working with a good coach on developing the proper mechanics for your body type (rather than just mimicking the mechanics of a good sprinter) is a critical, but often unappreciated aspect of training maximum speed.
Power Endurance: Power endurance is, as the name suggests, the ability to produce powerful movements repeatedly. Power endurance is incredibly taxing on the body's muscles, tendons, and nervous system because the muscles are operating above their lactate threshold and still required to produce maximal contractions. Middle distance runners, rock climbers, sprint swimmers, and cross-fit enthusiasts focus on developing power endurance. (... see cross-fitters, I'm showing you love...)
One of the limiting factors in power endurance is the bodies lactate threshold (the point at which lactic acid starts to accumulate in the blood stream). Below the lactate threshold, lactic acid that results from gylcolysis is effectively buffered, but at a certain point, hydrogen ions that result from hydrolysis within the muscle overwhelm bicarbonate buffers in the blood, resulting in acidification of the blood. In order to improve power endurance it is important to do interval training, where you have a "work" period near maximal levels followed by a "recovery" period in which you continue to work at very low percentages of your maximum. The details of effective and personalized interval training are too complex to give justice to here, but in general, interval training is an effective method of improving the body's lactate threshold and power endurance. Plus, interval training has the added benefit of improving endurance without the muscle catabolizing effects of high volume aerobic training.
Agility: Agility is the body's ability to change direction efficiently and is an emergent ability that results from the interaction of strength, speed, and coordination. Acceleration and deceleration are examples of agility. Agility is distinct from speed in that maximum speed requires no change in direction (i.e., the 100m is run purely straight ahead, tapping max speed, whereas the 30m shuttle requires several rapid changes in direction and continuous acceleration and deceleration, tapping agility). You can see the differential effects of agility and maximum speed in an elite 100m sprint. A more agile racer, with better acceleration is going to be faster out of the blocks, but a racer with a higher top speed will be moving faster as the race goes on. Thus, both acceleration and maximum speed are critical abilities to a sprinter.
Agility is required in just about all sports, but is perhaps most important in sports where the environment is either completely open or completely closed. An open environment means that the environment is undergoing continuous, unpatterned change. In open environments (such as combat sports, football, rugby, etc) agility is critical because an athlete must quickly and efficiently adapt to changes in their environment. In a closed environment there is no change in the environment other than changes initiated by the athlete (such as diving or gymnastics... imagine mounting a pommel horse to the back of a mechanical bull in a cowboy bar, suddenly shifting the environment from closed to open!). Thus, in closed environments, athletes can take advantage of the environment's stability to perform spectacular feats of agility.
A Few Training Prescriptions
It is important to recognize that athletes (beyond a young age) are not training for a general level of fitness, but are training to optimize certain biomotor abilities. There are several approaches to training optimization, but in general all of these training protocols rely on periodization. Periodization is a training program that emphasizes training different biomotor abilities throughout the year. Different sports follow different formats of periodization throughout the year based on (a) the needs of the sport and (b) the number of major competitions during the competitive season. Thoroughly understanding the needs of the sport allows coaches and athletes to preferentially train the most appropriate biomotor abilities and scheduling the training around major competitions helps to ensure that the athlete peaks at the correct time during the competitive season.
In general, periodization breaks the year (i.e., "the macrocycle") in to smaller training components ("meso-cycles"; such as the pre-season, competitive season, and off season). Within each of these meso-cycles different types of training are emphasized. For instance in the early pre-season athletes will spend timing developing a training base (anatomical adaptation to training) doing higher volume low intensity exercises, which helps to strengthen ligaments and tendons preventing injury later. As the pre-season progresses, the focus shifts to the development of maximum strength and maximum speed. Once the competitive season starts, the focus shifts based on the needs of the sport; power sports will focus on maintaining maximum power, whereas endurance sports will work muscular endurance of the appropriate duration. In the off season, coaches and athletes should encourage "active rest" and determine weaknesses or areas of improvement that the athlete can develop in the coming pre-season. This is a huge generalization, so I would encourage you to read this and this.
This type of progressive cycling throughout the macrocycle is referred to as "linear periodization" because the biomotor abilities being trained progress linearly from one meso-cycle to the next. An alternative and equally valid approach is nonlinear or undulating periodization, which posits the best way to optimize athletic performance is to shift the training focus within a "micro-cycle". That is, within a 10-day period some days should focus on power development, others on agility, and others on maximum strength with the number of days spent training a specific ability proportional to the needs of the sport (i.e., sprinters will have a lot of maximum speed and power days and few/no long endurance or agility days).
Personally, I like to hybridize the approaches when working with athletes and use linear periodization to shift the training focus on a larger scale (working on anatomical adaptation and maximum strength in the preseason and power in the competitive season) but within each meso-cycle I try to take advantage of nonlinear cycles. So within a meso-cycle focused on maximum strength development, on days when the athlete is feeling really good and fresh, we will train abilities with a larger neural component (such as power and speed). On days when the athlete maybe tired or just unmotivated, the focus will be abilities with a larger muscular component (such as endurance or strength). Thus, there is subtle variation within a meso-cycle, but there is even greater variation between meso-cycles.