I will get an invitation from a Romanian national while I am a third country national, I want to visit other parts of Europe during my stay there, is it possible to travel to other countries in Europe? is there any borders or any check points between lets say (Romania and Spain).

Thanks in advance.

Your permission to visit Romania is of no value in other EU countries. Either you get a Schengen visa or you get turned away at the border.

Between Romania and Spain in an indirect way intervene at least 6 countries and only 4 of them have common boundaries without checking.

It depends on your nationality. Some countries in Europe might require you to get a visa before you attempt to enter them. There's no one visa for all of Europe, you will need to figure out where you want to visit, and make enquiries at the Embassy/Consulate in question.

You need to be a member of the European Union for that. I'm sorry, but I don't think that's possible unless you manage to get the ID of a EU member