* This sign  [circumflex accent] used when it is necessary to  indicate long vowel [ long "a" in this case] in Arabic and Persian borrowings.

 * It is used to seperate the words which is written same but having different meanings. Such as ;

hala - aunt

Hâlâ - still,yet

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adet - number

Âdet - custom

[Aadet ]

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*  It is put on "a" and "u" letters if they followed by  light "g" and "k" sounds in some Arabic and Persian borrowings, such as ;

dergâh, gâvur, karargâh, tezgâh, yadigâr, Nigâr; dükkân, hikâye, kâfir, kâğıt, Hakkâri, Kâzım; gülgûn, merzengûş; mahkûm, mezkûr, sükûn, sükût

* If it is used for the purpose of  softening the sound , such as ;

kar = snow

Kâr = profit

then, you read  the consonant "K " softer.

Check this video about how the teacher reads  "â " [ at  1.14 minute ]

http://www.youtube.com/watch?v=T40k0SP_adk

  You have to learn those words, however there are some indications that gives us a clue about this kind of vowel dropping ;

* Some organ names with two syllables, when they take suffix starting with a vowel, their middle [second] sound drops.

burun ---> burun + um  ----> burnum

boyun ---> boyun + u   ----> boynu

ağız   ---> ağız + ımız  ----> ağzımız

alın    ---> alın + ın -----> alnın

karın  ---> karın + ım ----> karnım

beyin ---> beyin + i   ----> beyni

* When some two syllable words having wide vowels [a,e ,o ,ö] in their first syllable and narrow vowels  [ı ,i, u ,ü ] in their second syllable , take suffix starting with a vowel, the middle sound drops.

alın    ---> alın  + ı  ----> alnı

resim ----> resim + im ---> resmim

bağır  ---> bağır +  ım ---> bağrım

gönül ----> gönül  + üm ---> gönlüm

I took it as  " Selamun Aleykum " .I got it that way when I said there are some people saying it when they leave.

But if he meant just  " Selam"  than of course people don´t say just Selam when they leave.

Cus there is no chance for him ever heard  Selam used when two Turks leaving eachother.

http://www.radikal.com.tr/yazarlar/cuneyt_ozdemir/bir_iki_ucler_yasasin_turkler-1109271

As Gokuyum said,  it is racist, meaningless thing that was made up by some racist minded people who tried to spead it among childeren. I heard that unfortunately, I remember when I was child, some kids were singing this thing and it sounded fun to us in those times  but now we realize  how evil it was to spread such things among children and washing little brains with hatred against other nations.

I hope there is no child still remembering this song.

Below, some more examples :

ağız / ağzı

alın / alnı

bağır / bağrım 

beniz / benzi

beyin / beynimiz

boyun / boynu 

böğür / böğrüm 

burun / burnu 

geniz / genzi

göğüs / göğsün

gönül / gönlünüz

karın / karnı 

oğul / oğlu

Some words [especially loanwords ] lose their final vowel (apocopate) when a suffix which itself begins with a vowel is added to the noun.

Example:

izin - leave, time off - becomes izn-im [NOT izin-im]- my leave 

in the example, the final vowel of the noun root is dropped when adding a suffix which begins in a vowel.

-fikir

fikrimiz - our idea

- keyif
keyfi - his/her joy

oğul
oğlum - my son

boyun
boynu - his neck

nakil
nakli - his transport

ahit
ahdi - his promise 

* In the last two examples above that the final consonant -t has changed into its soft form -d.

kayıp  - loss

kaybı - his loss -

* in the example above, -p  softens into  -b when it takes possesive suffix.

In meaning there is no difference, the only difference is emphasizing [strengthening] the negative or interregotive statements.

Istanbul´a tatile gitmiyorum.  ---> I don´t go to İstanbul for holiday.

Istanbul´a tatile gitmiyorum ki. ---> I don´t  go to İstanbul for holiday.

Hayat  ne ki sonuçta ?  What is life all about ?

[At the end of the day what is life all about ?]

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ki  as a strengthening particle

-ki  can be used to strengthen the question and negative statements.

-Beni istemiyormuş. Ne yapabilirim ki ? 

-He/She doesn´t want me. What can I do ?

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Davet etmezsen gelmez ki ! 

He/She would not come if you don´t invite him/her !

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Acaba rica etsen bu işi yapar mı ki ? 

Will [would] he/she do this job if you ask [politely] him/her ?

Adjectives can  be intensified  by  adding  * presound [ first two letters of the adjective ] plus  -p, -m, -s , -r  in between.

First two letters  of  adjective  +  -p, -m, -s , -r    +  adjective

 ZO                           +  P       +  ZOR  ----> ZOPZOR [ very difficult ]

 DE                           +   P       +  DERİN  --->  DEPDERİN [ very deep ]

 KA                            +  P      +  KARA  ----->  KAPKARA  [very dark ]

 BO                          +  M       +  BOŞ  ---->  BOMBOŞ  [ very empty ]

 BE                           +   M      +  BEYAZ ---->   BEMBEYAZ  [ snow white ]

 Sİ                            +    M      +  SİYAH  --->  SİMSİYAH [ Extremely black ]

 BE                      +  S       +  BELLİ   ---->  BESBELLİ [ very obvious ]

 PE                       +  S      +  PEMBE  ---->  PESPEMBE [ very pink ]

 BÜ                       +  S     +  BÜYÜK -----> BÜSBÜYÜK [ very big  ]

TE                       +  R     +  TEMİZ  --->  TERTEMİZ [ very clean ]

SE                       + R     +  SEFİL  ----> SERSEFİL [ very miserable ]

* If the adjective starts with a vowel, then only first letter is used.

A                   +    P    +  AÇIK  ---->  APAÇIK  [  very obvious ]

U                    +   P     +  UZUN ----> UPUZUN  [ very long ]

Although it is not commonly used, there are some people saying it to eachother when they leave , as in the meaning of  " Bye ", however  it has deeper meaning which is " May God´s peace be upon you ". Obviously ,it is religious way of saying " bye " [be in peace]