Be sensitive to people's sensitivities and be insensitive to people's insensitivities
First I'd like to give a couple of definitions from dictionary.com:
1. endowed with sensation; having perception through the senses.
2. readily or excessively affected by external agencies or influences.
3. having acute mental or emotional sensibility; aware of and responsive to the feelings of others.
1. deficient in human sensibility, acuteness of feeling, or consideration; unfeeling; callous: an insensitive person.
2. not physically sensitive: insensitive skin.
3. not affected by physical or chemical agencies or influences: insensitive to light.
Now I'd like to focus on the first part
Be sensitive to people's sensitivities
By this I mean do your best to be aware of what others are sensitive to and be sensitive to that. Someone who is overweight, someone who is excessively tall or excessively short, someone who has just had a family member die or put in prison or ostracized - these are all examples of people who may have a sensitivity to something, and so you should be sensitive to it - ie, do not go into those topics with those people unless it is a truly heart to heart conversation. In a group setting, do your best to steer conversation away from the topic that is sensitive to one of the people in the group. By being sensitive like this you help to protect the person who is sensitive, you create trust with that person, and you hopefully teach others to also be sensitive.
Be insensitive to people's insensitivities
We've all heard the idiom 'Like water off a duck's back' meaning a person is impervious to any criticism and are not affected by it in the slightest. So let us say that you are overweight, excessively short or have some kind of minor physical mark like a poorly located scar. People who make insensitive remarks towards you, be insensitive to that, ie, react to it like water off a duck's back. By being insensitive to situations that warrant it, you will help to decrease your own stress level and allow you to live a happier life. A by-product of this could also be a decrease in the amount of insensitivity that comes your way.
Putting this into practice
Now obviously behaving like this is more of a challenge than simply to read or write it. It is also quite a mouthful if you want to use be sensitive to people's sensitivities and be insensitive to people's insensitivities as a mantra. However, reading this phrase on a regular basis will feed it into your subconscious and help to guide you in the right direction to start to think in these terms and behave in this way.
I started thinking in these terms when I was in university when I would see and be a part of situations where other students were being carelessly treated. I have continued to think in these terms up to now and it has helped me, I believe, to be a better person. Of course, there is always room for improvement.