To 'file' means that an attorney or an individual take documents to the court in order to pay the fee and get the suit placed on the court's docket. It could be the filing of the initial petition (or complaint), an answer, a motion, or other documents related to the lawsuit.
A file is also a group of documents that is related to each other because of subject matter, usually a person. A file contains all of the facts and copies of documents used in the given subject so as to give rise to the facts at hand if ever required. In a legal sense, a file on a person contains the legal matter which the person is using the law firm for and copies of all the evidence against or for the person. The type of evidence differs in files, but the cumulative and comprehensive nature of file is that everything can be found in one place.
To file documents with the court, the attorney or the individual would go and see the court clerk. From there, it may also be required to also see the Judge's assistant. If this is a necessity, the court clerk will tell you to take copies of the documents to the office of the assigned judge.
A legal professional, just like all other professional service providers, will maintain individual files on each client they have in order to ensure that as the facts grow the total subject is added to and none of the facts are forgotten.