Ariza is basically a du`a which, instead of saying, you write it down on a piece of paper. There are ahadiths where the Imams have told us that we can write down our prayer (du`a) in a paper and put it in a well or a river. See for example, a similar hadith in the last section of Mafatihu 'l-Jinan under the section "ruq`a-e hajat".
ARIZA
Meaning of ARIZA in English
Advanced Islamic English dictionary. Расширенный исламский словарь английского языка. 2012