A person has integrity if their thoughts, speech, and their actions are constantly being more rather than less aligned. One may have both hypocrisy (failing to live up to one's own standards) and integrity, though many failures without changing one's approach would signal a lack of integrity not just a lack of success.
An individual who is so aligned and has sucessful long term relationships with others beings of integrity, can be said to be a person of integrity. It is a very simple concept that is very difficult to implement.
The benefits of integrity are well known: such people can commit and deliver on long term goals even if there is controversy over tactics, thus they can be trusted by their peers, and are dangerous only to their avowed enemies: they keep promises when promises to them are kept (and go to great lengths to cover when they are left hanging), they are mindful of their reputation with others of integrity (though may sometimes be unconcerned with reputation with those who lack it - to please your enemies is to fail), act with honour at least on matters important to their peers, and are in general worth dealing with even if they are difficult, because they mean what they say, and say what they perceive to be real.
For civics and communities integrity has sometimes been represented as having three elements basic to cooperation:
1. Being truthful. - That one gives an honest answer to an honest question, or at least a substantially similar question that resolves the same dilemma or enbles the same choice (not every honest question should be answered)
2. Being true to one's word. - That an individual should make and keep their promises, and should take some pains to ensure that the conditions and constraints on the promise are understood by those who receive it. For instance when promising something to children it is important to be very clear.
3. Being true to oneself. - An individual who declares personally their code of ethics, their philosophy and their values has relatively more integrity than one who doesn't, or who constantly changes these.
A person of integrity is then someone who achieves all of these, most of the time, and is honest about any lapses in their conduct. For instance, issuing apology when wrong, whether they were caught by others or not.