Iltizaam (clinging) is when the supplicant (person making du’aa’) places his chest, face, forearms and palms against it and calls upon Allaah saying whatever du’aa’ he wishes. And this is done by clinging to the part of the Ka’bah that is between the Black Stone and the door of the Ka’bah.
There is no specific du’aa’ that the Muslim should say in that place. He can cling to the Multazam when he enters the Ka’bah (if it is easy for him to enter) or he may do that before performing the Farewell tawaaf (tawaaf al-wadaa’), or he may do it at any time he wants. He should not cause difficulty for other people by offering a lengthy du’aa’. Similarly it is not permissible to crowd other people or annoy them in order to cling there. If he sees a space then he should say du’aa’. Otherwise it is sufficient for him to say du’aa’ whilst circumambulating. 
Yahya related to me from Malik that he had heard that Abdullah ibn Abbas used to say that the area between the corner of the Black Stone and the door of the Kaba was called al-Multazam. Muwatta Malik Book 20, Hadith 957