Whenever you do a workout you should stretch all major muscle groups after your cooldown while your muscles are still warm and flexible. Some schools of thought also believe that you should do a quick stretch after your warmup as well, but the jury is out on that one; if you want to stretch following your warmup make it reasonably quick, don’t push it too far and don’t bounce, as your muscles won’t yet be fully warmed up, and stretching a cold muscle doesn’t do it any good!
I believe post-exercise stretching is just as important as warmup and cool down as over time it will improve the range of movement in all of your joints, which will improve your general flexibility. How flexible you are is usually down to the individual; some people are naturally more flexible than others and those who have trained as dancers or gymnasts are hyper flexible – in other words they have too much flexibility in the joints, resulting in lengthened ligaments and the risk of problems with joint stability in later life.
Others just seem to be naturally stiff; footballers and horse riders for example famously have short, tight hamstring due to their activity and positioning. Yes, their hamstrings will be good and strong but they should specifically work on lengthening them as short, tight hamstrings can pull the pelvis downwards thus causing strain on the lower back.
Inflexible muscles are more likely to be injured when asked to perform an extreme or accelerative movement, such as a spurt or a speedy change of direction during sport. Ideally, training for these sports should include exercises that take muscles through the full range of movement, thus keeping the muscle the correct length, and scientific research has proved that more flexible athletes perform better at their particular sport.
For the non-athlete merely working out in the gym, stretching is just as important as it keeps the body balanced, reduces strain and also feels very good, thus promoting a feeling of well-being following a workout.