What are benefits of author's personal copy of article and how I can order it? I have some publications in Elsevier and Springer journals. After publication, I received an author's personal copy of one of my articles. I don't know how they sent it to me because I don't remember ever had requesting a personal copy. After that I did not receive any personal copy for any of my published articles. Lately, I had a conversation with my supervisor and he asked me to provide him the author's personal copies of my published papers as I was the communicating author. He told me that the author's personal copy is mandatory for official use.

I could not understand that. How could the author's personal copy be anyway different from original copy for the purpose of official use?
I think you are hitting legacy procedures. Before the internet and electronic publication and such, author copies were very usual. Since the publisher was busy typesetting the copy anyway, they would print off some number of copies of the paper to give to the author. Depending on the journal, these might even be typeset in exactly the way they were in the journal, starting at the same place on the page for example. Then when researchers wanted copies they would request them from the author. 

It was usually cheaper, and higher quality, than photocopying. It also acted as a nice intro between the author(s) and people interested in the work. The prof I did my post-doc under had a huge collection of author copies. 

These days, electronic publication has mostly taken over. Protected PDF copies, with built in water marks, for example, are fairly common. And usually a lot cheaper and more convenient. You don't have to wait for mail just download. And you can use them on your favorite device. They're searchable, etc.