Then again, lets see what warming up is defined by. Warming up basically raising your body temperature. Why would you want to do that? Increasing your body temperature allows your muscles to flex and stretch more, obviously, since cold muscles don't like to move freely. Your tendons, ligaments and cartilage also increase in flexibility, leading to a lesser risk of injuries, strains and tears.

A good enough warm up should get your heart pumping and a little bit of sweat. A basic warm up would be walking slightly faster than average for 10 minutes before jogging. Your heart will be ready to accept the need for blood and nourishments of your muscles.
Warming up reduces pre-mature fatigue, maybe you haven't been warming up properly and you're a better runner than you think? It also enables you to get the most benefits out of a workout.
The lenght of the warm up should be dictated by the intensity of the following exercise. As I said, average jogging should be prepared for with 10 minutes of walking slightly faster than average. Don't have the time to warm up? Get ready for severe injuries and a heart attack. Cold muscles do not want to stretch and work.
If you're interested in my warm up - it would take quite a while to write it all down. But it takes just around 15 minutes, consisting many different ways to warm up, like Jumping Jacks, Scorpions, Trunk Twists, Clap push-ups, Lat pulldowns and so on and on. I do 18 different exercises to warm up, so as you can see it can get quite complex. There are also ways to fix your muscle anatomy with warming up in specific ways before a workout.
Always make sure to properly warm up before exercise, as it can increase your results, help the recovery be more bearable ( keep in mind pain is not the only indicator of a good workout ) and prevent injury.
If you have no idea how to start running, here's a basic week:
Drink a lot of water, do not underestimate sleeping properly, stop making common mistakes and you'll be much happier!