Public domain is the status of any recorded work that’s copyright or patent has expired or that never had such protection. Nursery rhymes are examples of public domain
Fair use is a legal doctrine stating that portions of copyrighted materials may be used without permission of the copyright owner provided the use is fair and reasonable, does not substantially impair the value of the materials, and does not curtail the profits reasonably expected by the owner. An example of fair use is researching a topic online for an essay. This is ok, as long as your references are properly cited. If not it will lead in to you plagiarizing someone else's work as if it was yours.