Haya is an Arabic word derived from the word hayat, which means life. This term covers a wide number of concepts. In English, it may be translated as modesty, shyness, self-respect, bashfulness, shame, honor, humility, etc. The original meaning of haya refers to a bad and uneasy feeling accompanied by embarrassment, caused by one's fear of being exposed for some unworthy or indecent conduct.  
Haya in general means natural or inherent shyness and a sense of modesty.
Haya is an attribute which encourages believers to avoid anything objectionable. 
Indeed, Allah is not timid(Shy) to present an example - that of a mosquito or what is smaller than it. And those who have believed know that it is the truth from their Lord. But as for those who disbelieve, they say, "What did Allah intend by this as an example?" He misleads many thereby and guides many thereby. And He misleads not except the defiantly disobedient. Quran Surah Baqrah 2:26
Then one of the two women came to him walking with shyness. She said, "Indeed, my father invites you that he may reward you for having watered for us." So when he came to him and related to him the story, he said, "Fear not. You have escaped from the wrongdoing people." Quran surah Qasas 28: 25  
Narrated by Abu Huraira (R): The Prophet Muhammad (May Allah honour Him and grant Him peace) said, "Faith (Belief) consists of more than sixty branches (i.e. parts). And Haya is a part of faith." Sahih Al Bhukari Vol.1:9
Modesty is an ingredient of Iman (faith). Sahih Muslim 36
The Prophet (صلى الله عليه و سلم) was more shy (from Haya': pious shyness from committing religeous indiscretions) than a veiled virgin girl. Sahih al-Bukhari 6119 (Vol.8: 140)
Narrated Abdullah ibn Umar (R): The Prophet Muhammad (May Allah honour Him and grant Him peace) passed by a man who was admonishing his brother regarding Haya and was saying, "You are very shy, and I am afraid that might harm you." On that, Allah's Apostle (May Allah honour Him and grant Him peace) said, "Leave him, for Haya is (a part) of Faith." Sahih Al Bhukari 6118 (Vol.8: 139)
The above hadith is also a form of proof that "shyness" is not just something regarding women but also an attribute that believing men should have, for it is an indication of their fear of Allah (Glory be to Him) and an indication of the value of their Deen (way of Life as prescribed by Allah (Glory be to Him)).
We often find that shyness, humility and bashfulness are frowned upon by our society as a weakness or a lack of confidence when, in fact, these are qualities of a dignified upright human being, who is conscious of his actions and his responsibilities in life. 
While modest behavior varies from person to person, guidelines do exist in the Qur’an and the Sunnah. The code of modesty for both men and women includes:
· Lowering the gaze and avoiding flirtatious speech and conduct;
· Avoiding close physical contact with unrelated members of the opposite sex;
· Observing modest or Islamic dress according to the Qur’an and Sunnah;
· Not drawing unnecessary attention to oneself.
'Allah, the Mighty and Sublime, is forbearing, modest and concealing, and He loves modesty and concealment...'"Sunan an-Nasa'i Vol. 1:406