Stretching is something that virtually everyone knows is important, and knows the basics of, but rarely is it ever done enough. Coming from a long gymnastics background, I suppose I am an exception to the average level of flexibility, but I do believe that a person who does not stretch as extensively as I used to can still be flexible. Stretching does come with a few tips:
Stretch when you are warm. This is so important! When I was in gymnastics, my coach would have us gymnasts run before we stretched, and I didn’t exactly know why. I never paid attention to the difference between when I stretched when my body was cold and when it was warm. Now, as I’ve aged and my muscles have lost some of their strength and flexibility, I am a bit more aware of what works and doesn’t work. When I coach now, it’s difficult to stretch with the kids if I’m not warm myself, so my first piece of advice is to make sure that your muscles are as warmed up as possible before you begin stretching. If your muscles are 
cold, then they are also tense, and the point of stretching is to relax and elongate the muscles. I am most flexible after I exercise and when I am in a hot tub or a sauna, which I know every other person who stretches regularly would agree with.
A typical stretching routine calls for holding each stretch for 10-30 seconds. This is what I have my gymnasts do when I coach, and when I was in gymnastics we would often hold the stretch for closer to 45 seconds. The 10-30 second rule has been around for a long time, but new suggestions are arising that stretches should be less intense. When I stretch on my own, I usually breathe in, then go into the stretch while breathing out and only hold for about 5 seconds. Then I return to normal, breathe in, and hold the stretch again while breathing out. I will repeat this 3-5 times. Breathing out while going into the stretch I have noticed does make a difference, and I used to never believe that breathing at certain points of stretching or exercise was important. This type of stretching has proven less painful for me than the common 10-30 hold, so I urge others to try it as well.
One last thing, make sure that you stick with stretching! I hear a lot of complaints from a variety of people about not being flexible, but hardly anyone takes actions. There are always going to be those people who are born a little more flexible than you, but I think that regular stretching, on an every-other-day basis at the least, will yield promising results.