Time under tension - TUT as its known is when you place the muscle being worked under tension for a period of time during your reps/sets.
Now it's said that our bodies are constantly under tension from gravity on a daily basis. When we perform any lift we are already under tension. This is true. Others will say that the duration we spend performing those reps in a set is also time under tension, for example 12 reps will take longer to complete then say 5 reps.
However actually slowing down the repetitions themselves is what we will talk about here. There are many variations but the main one we will discuss is the eccentric portion of the lift. This is the part in which the muscle is being stretched (the downward phase of a squat, as you lower yourself from a chin up, lowering the bar to your chest on the bench etc) This part of the lift induces the most damage on the muscle. With this damage comes adaptation to grow and repair. This costs the body energy to do so.
Recent studies have suggested that there is a value to manipulating your tempo when you train. The results showed that it burns more energy during and especially post workout. Greater post exercise energy consumption means your metabolism is elevated, therefor more calories may be burned in the hours preceding.
I myself have trained using this method and used it on many clients with great success. However is it not for everyone. Performance athletes will want to avoid too much damage (DOMS) as their frequency of training and the ability to recover is of a much higher need. They also at times need speed/velocity and so slowing down the lift may be counterproductive.
Some people may also find this way of training not very stimulating mentally and may get bored of the slow tempos and feel that it may drag a session out, it does come down to preference, if you wish to speed things up just remember to always be in control of the weight at all times.
Personally i think it's great for any newbies or people who may be training with old injuries as It allows you to master technique, and get a very demanding workout done without placing the connective tissues under too much stress. It can also aid strength or break any plateaus as i've seen with people who return to lifting with no tempo push through the weights with ease, from using this method.