The word 'paper' is American English. In British English we say 'essay'. Papers or essays are generally related to undergraduate degrees. The word 'thesis' is generally associated with the final piece of writing for a PhD student. A 'dissertation' is for a master's degree.
However, in the United States the reverse is true. The final doctorate paper is called a dissertation and the master's degree requires a thesis.
Yang Li took up a question by Minken and explains the difference between the words paper, dissertation and thesis.