The meaning of hayaa is the restraining of oneself from an action for fear of doing something blameworthy.  According to the hadith of the Prophet (sas), hayaa is a praiseworthy attribute.

"An Ibn Umar anna an-nabiyya marra 'alaa rajulin we huwa yu'aatibu akhaahu fiy al-hayaa yaqoolu:  innaka latastahyiy.  Ka'annahu yaqoolu qad adharra bika fa qaala rasoolu allahi: "Da'hu fa'inna al-hayaa min al-iman."
On the authority of Ibn Umar:  The Prophet (sas) came upon a man who was criticizing his brother for hayaa, saying:  'You are so modest (or so shy).'   as if he meant to say that this has harmed him.  The Prophet (sas) said to him:  'Leave him, for hayaa is from faith.'. Al-Bukhari

The Prophet (sas) defined hayaa as a praiseworthy attribute, period.   Linguistically, the word hayaa could include a type of modesty or shyness which is not praiseworthy such as modesty which prevents one from enjoining right or forbidding wrong or modesty which prevents one from asking a question about something which they need to know.  This, as in the popular statements:  "There is no modesty in religion."  or "Knowledge get lost from two causes:   arrogance and modesty."

The modesty needed by a Muslim is of three types:

1. Modesty before Allah Most High.  This is when one experiences an urge or an inclination to do something disliked by Allah.  Its result is the witholding from committing the forbidden and the consistent fulfilling of the obligatory.
2. Modesty before people.  This consists of treating them fairly, witholding one's harm from them and not showing them any ugly side.
3. Modesty with oneself.  This results in obeying Allah just as much in private and when no one else is around as one does in public.

The Prophet (sas) said:  "One part of that which has reached the people from the first revelations is:  If you have no shame, do as you wish."

This hadith is part of jawaami' al-kalim (the ultimate in eloquence) which is one of the special gifts given only to Prophet (sas) Muhammad.  The others were fear struck into the hearts of the enemy when the believers are still one months march away and that the entire earth is considered clean and a place where prayer can be made.  This hadith has several different meanings, all of which are correct.

1. That the sentence is a command (as it indeed appears in the language), but with the meaning of condemnation and threat.  So the meaning is:  If you have no shame, then do whatever you wish and Allah will repay you fully for all that you do!  This is quite similar to several verses in the Qur'an and some hadith:
   
   {I'maloo maa shi'tum innahu bima ta'maloona baseer}
   {Do as you wish. Surely, He sees all that you do.}  Fusilat: 40
   
   {Fa'badoo ma shi'tum min doonihi. Qul inna al-khaasireena alladhina khasiroo anfusahum wa ahleehim yauma al-qiyama alaa dhalika huwa al-khusraan al-mubeen}
   {So, worship whatever you want beside Him.  Say: Those in loss are those who lose themselves and their families on the day of Qiyama.  Verily, that is the clearest of loss.}  Az-Zumar: 15
   
   The Prophet's (sas) statement: "Whoever sells alcohol, let him cut pigs."  (Ahmad & Abu Daud, but its chain is weak)
2. The sentence is a command, but with the meaning of information. And so the meaning is:   Whoever has no shame will do whatever they wish since the main barrier to committing the obscene and the wicked is shame.
   
   Ibn Abbas said:  "Modesty and faith are like two horns.  If one is removed, the other one follows it."
3. The sentence is a command and the meaning is a command.  The meaning then becomes:   If a particular statement or act causes you no shame, then go ahead and do it.   Again, meaning that shame or modesty is the main preventative of evil and a good measuring stick.
   
   As the Prophet (sas) said: 
   "Al-ithm ma haaka fiy sadruka wa karihta an yattila'a 'alaihi an-naas."
   "Sin is that which disturbs your heart and which you would hate for people to see."

The Prophet (sas) said:

"Al-hayaa laa ha'tiy illa bikhairin"
"Modesty brings nothing but good."  Muslim & Bukhari

"Al-hayaa shu'batun min al-iman"
"Modesty is one portion of belief."  Muslim & Bukhari

Modesty can either be a natural component of ones character, or it can be "earned" or "acquired" in Islam by learning, self-discipline and effort.  Thus one who was lacking in this character before Islam will have to work harder to acquire it in Islam.

The Prophet (sas) said to Al-Ashajj Al-Asriy:  "You have two character traits which Allah loves."  He said: What are they?  The Prophet (sas) said: "Restraint and modesty (al-hilm wa al-hayaa)."   He said:  All along, or just recently?  The Prophet (sas) said:   "No, all along."  The man said:  Al-hamdu lillahi who placed in me two character traits which He loves."  (Ahmad and others and its narrators are all people of the sahih and similar narration in Muslim without the question about "all along or just recently")
And so, modesty is these two types:

1. That which is inherently in one's nature and one did not work to acquire it.
   
   Umar said:  "Whoever is modest hides and whoever hides fears Allah and whoever fears Allah is protected."
   
   Al-Jarrah ibn Abdullah Al-Hakamiy said:  "I stayed away from sin for forty years out of modesty, then I found piety."
2. That which is acquired by means of knowledge of Allah.  This is the greater of the two because it is built on a foundation of knowledge and consciousness of Allah.
   
   The Prophet (sas) said to a man:  "Be modest with Allah as the family members are around the best of them."

True modesty - especially the one before Allah Most High - is not a mere feeling or issue of character.  It has concrete results:  it leads one to fear and obey Allah in all things large and small, public and private.

The Prophet(sas) said: "Be modest before Allah as is befitting Him."   They said:  We ARE modest before Allah, al-hamdu lillahi."  The Prophet (sas) said:  "That's not what I mean.  The one who is modest before Allah as is befitting Him, let him guard his head from what it contains and his stomach from what it takes in and let him remember death and destruction.  And whoever desires the hereafter, leaves the ornaments of this life and whoever does all that has become modest before Allah as is befitting Him."  At-Tirmidhi