Service of process is when a party involved in a lawsuit (Usually the Plantiff) provides a notice of legal action to the other party involved (usually the Defendant), by the Court or other entity to allow that person to respond to the lawsuit.
Process Serving is taking the documents from the one party to another. For Example, Mr. Landlord is suing Mr. Renter for failure to pay rent. Mr. Landlord gets the appropriate documents ordered, contacts a Process Server (Who are legally authorized to deliver the documents) gives the documents to the Process Server , who in turn finds and delivers the documents to Mr. Renter notifying him that he is being sued for failure to pay rent.
Process Servers must follow specific guidelines. Each state is different. Once a serve has been made, a server will validate and sign an affidavit that the serve has been completed.