Is it possible for active people to truly live practice "mindfulness"? Several buddhist/meditation books recommend me to practice "Mindfulness" or living in the present moment alone.

I am a married man with a 4 yr old son and wife.I am currently working in one of the top 10 IT company in the world.

On any given time I will have a "list of things to do" which are important and which are pending from my side.

So if I ignore my tasks and "live only in the present moment" it will greatly harm me and my family's life.
Also how am I supposed to be at peace when all the pending items keep reminding me that they are not yet completed even if I try to calm myself for couple of minutes.

Does this mean that people with lot of responsibilities should refrain from practicing "mindfulness" ?
Mindfulness or living in the present moment is not the same as not having goals or responsibilities. Instead it is about doing everything you do with full attention and without judgement.

For example, an active and not very mindful person would drive a car, look at his watch to see what time it is, and nervously think about everything he still has to do today, all three at the same time. A mindful person would drive the car with full attention. When other thoughts pop-up he would accept it and let them just pass by without giving them too much attention. Only when he has arrived at his destination would he actively think about his next activity.

This means that you still can have goals, it's just that you are not (in your mind) busy trying to achieve multiple goals at the same time. Additionally, being mindful also means that you accept whatever happens without judgement. If you accidentally hit another car while driving, you (try to) don't get angry or upset, but instead accept and deal with the consequences of the accident.