What is istijmaar (lit. cleaning oneself with stones after eliminating urine or stools)? Is it proper istijmaar to clean oneself with tissues and water when on board a plane? What is the proper way to cleanse oneself? 

Praise be to Allaah. 

Istijmaar means to cleanse
the front or back passage of urine or stools using stones or something
else in their place. One of the things that may take their place is
tissue, but on the condition that one wipes no less than three times.
And one should not use things that are forbidden to be used for istijmaar
such as dung and bones, or things that have a kind of sanctity such
as food etc. 

It is permissible to do istijmaar
with or without water. The scholars said, it is better to use both,
because that is cleaner.