Great question. Before you can understand the answer, you need to understand some of the principles involved in how "tempo" relates to performance. Performance relating to either hypertrophy and endurance or power/strength.
It is generally well known and accepted that to invoke proper hypertrophy, you need to introduce enough stimulus to the muscle to break it down, so that it can rebuild itself via protein synthesis later. Now, how do we actually do this? Well, no one REALLY knows, but from various research papers, journals, studies, and just experience, it seems that time under tension (TUT) is a key factor in this, as it stimulates the mTOR pathways which are also key in building muscle (hypertrophy).
Now that we know we need time under tension to invoke hypertrophy, we need to come up with a tempo to achieve this in a systematic matter. Now, it is VERY IMPORTANT that you understand there is no ONE right tempo. In fact, you should NOT use only one tempo for all your hypertrophy goals (or any goals) as your body will adapt and you will halt your progress. Moreover, you will miss out on the benefits of other tempos, miss out on different muscle stimuli, as well as cause muscle imbalances in the long run.
That being said, there are definitely some tempos that are better suited for hypertrophy rather than power, since they are achieved differently. For example, since hypertrophy (and endurance) require greater time under tension, you will spend more time in the repetition than for power/strength movements. A very good tempo for this goal would be to explosively lift the weight up, so 0.5-1 second lift, 1-2 second hold, and 2-3 lowering the weight. It is important that you understand why I chose these numbers. First, throughout many studies it has been established that the concentric part of the movement (lifting it, i.e pressing, curling up, squatting up) stimulates the most muscle fibres through explosive movement, I can reference many articles for this but a quick google search will show the same answer. It is NOT true that you should lift the weight up slowly, DO NOT DO THIS, you want to lift the weight up as fast as possible, explosively, with proper form assuming you warmed up sufficiently. Now, the isometric part ( the static part in the middle of the lift), you should always hold for at least one second in order to invoke your stabilizer muscles and get a good blood flow for nitric oxide transportation to your muscles to deliver nutrients to them. However, this part shouldn't be too much longer than a second, since we are trying to maximize TUT and you will be resting your muscles in this position, taking away tension. Now, the part of the lift, that breaks down the most muscle fibres and therefore influences the greatest size gains, is the lowering portion (eccentric). This is the part where your muscle is lengthening itself, and therefore the more time you spend doing this, the greater this effect will be, and thus a minimum of 2 seconds and max of about 4 is ideal since this is very stressful on the muscles and you don't want to overdo it, or you won't recover well. So that's hypertrophy, right ? No...not really.
Remember how I said time under tension is what matters? Well, the above portion takes care of the time part of the equation. But what about the actual tension, i.e the weight? Well you can also invoke a slightly different form of hypertrophy (there's two, one being volume based with high reps, and one density based with low reps heavier weights). To do this, you just lift heavier weights for lower reps (but not too low). So for example, if you do 3-4 sets of 8-12 reps for the first form of hypertrophy, you could do 5-10 sets of 3-6 reps with heavier weights for the second form. Tempo wise, you COULD apply the same tempo but it is not advisable, since with heavier weights, the eccentric part of the movement will be very hard on your muscles if you do it for longer than 2 seconds. So a good tempo for example could just be 1-1-1 or 1-2-1 etc...
Now, for power and strength , you must first understand what they are. Strength is how much weight you can move and is mostly determined by your CNS, i.e your neural ability to lift weights, the density of your muscles, and form. I'm not an expert but I am a national level powerlifter for Canada so I can help out a bit with this. Power, on the other hand, is how fast you can apply that strength over time, think "move weight fast" as opposed to "move weight" for power vs strength. Now, from the way we constructed the above analogy, you might already know how to structure tempo for power... Yup, FAST. As a matter of fact, most explosive athletes like olympic powerlifters and football players will completely skip the eccentric part of the movement (the lowering) so they can spend more time on the other two parts which will maximize power, neural efficiency, and form. Since you're probably not either of these, it is sufficient to just minimize time in the last part of the lift so you can focus more on developing power, so don't spend more than a second lowering the weight. This means, if you are trying to improve your power, for the bench press let's say, you would pick a weight that you can move fast (50-60% of your 1 REP MAX), unrack the weight, lower it for no longer than a second until it touches your chest, pause at least a second or two at the bottom (isometric) in order to make sure you don't have any rebound, and PUSH up and explode as fast humanly possible with good form (should ideally take less than a second)...that is more or less how you improve power, and train for power. So the tempo would be around 0.5 seconds lift, 1-2 seconds hold, 1 second down.
Strength isn't much different, but you must understand for strength, that the main factors are your form and your neural efficiency. Your neural efficiency is basically how well your CNS is able to recruit muscle fibres under heavy loads. The only way to improve this efficiency, is to do a lot work in the lifts you want to get stronger at. I.e if you want a stronger squat, you must squat A LOT, as in sets, reps, overall work over time. Generally, you would want anywhere from 5-10 sets of 1-6 reps for this purpose. The tempo, accordingly, will ALWAYS (specially for power and strength) be as fast as you can up, i.e max 1 second lift, I suggest 1-2 second isometric hold (middle of movement) to strengthen your connective tissues and tendons for more strength work, and no longer than 2 seconds eccentric (down) since heavy weights will take their toll on their body if you spend too much time under tension.
So, assuming your form is good, what is the key to all of this? For hypertrophy and endurance, we want to break down as much muscle as possible, this means lowering the weight for a longer time (i.e 2-4 seconds) on the way down of a curl, or descending the bar into your chest in bench, or lowering yourself in the squat. The amount of reps you do will determine hypertrophy and endurance. 6-12 reps is generally better for hypertrophy, while anything plus 15 reps will always be endurance based.
For strength and power, since you will be working with relatively heavier weights, the weight is what takes care of the tension and thus you won't need to (or should) lower the weight for longer than a second or two at most, as you'll get injured and it ll be hard to recover in general if you don't.
What's common to all tempos? The beginning portion of the lift, the concentric phase. It is basically never beneficial to spend a long time, i.e longer than 2 seconds to lift the weight up, since lifting it faster breaks down more muscle, recruits more fibres, and also, training your muscle to move weight up slowly is generally a horrible idea. They will become adapted to that speed, and as a result, any relatively explosive movement will be more likely to cause injury. Please don't train your muscles to be slow.
Hope that helps.