Awaara [Hindi:आवारा,Urdu:آوارہ] is a Hindi/ Urdu word which means someone away from or without a family, someone who roams around without any work etc., a wanderer, vagrant.
Awaargi or Aawaargi is the word for such state of wandering, vagrancy. The word can be heard in many songs including the one titled 'Main aur Meri Awargi' (I and My Vagrancy) sung by Kishore Kumar and then Nusrat Fateh Ali Khan. Awargi is also the title of a famous ghazal sung by Ghulam Ali.
Awarapan would be a word with the same meaning as Awargi. As such, the word didn't really exist or at least used much until the movie Awarapan came up, as Awargi was the used form for the word. However, like one can add -ness to any adjective and create a noun, similarly in Hindi one can add -pan to many words and make nouns out of adjectives, like paagalpan from paagal. So here they have added -pan to Awara and made it awarapan. Wandererness, or Vagrantness if you will, or simply, Vagrancy.