Integrity is a little bit like Truthfulness combined with Morality. Integrity is not really a common word in English. A lot of people will never ever use this word in their vocabulary at all. It's a sense of a feeling of straightforwardness, we are honest, we are truthful, we have high moral ethics.
To have integrity means you treat people fairly, you treat people kindly. You treat people in a way that you would like to be treated. As a typical expression, "Do on to others as you wish they do on to you."
Having integrity means you stand up for things that are good, i.e. you try to stop things that are harmful, as well. It has a lot to do with Compassion. When we think of a person with high integrity, it means they keep to high moral ethics, keep to good standards, they try not harm others. They are people you can trust. They are also people who are capable of becoming a leader. So integrity has a lot of good qualities in it.