“Process” is a term referring to judicial paperwork which directs a person or company to do an act according to the rules of the court. A Summons, Subpoena and Order to Show Cause are common examples of process. Process needs to be given to that person or company in a manner specified by the laws of the court in that location, this is called service of process. Service of process must be performed according to the local rules to ensure that the party receives notice of the courts order, which in turn is a prerequisite to the courts enforcement proceeding. If proper service is not made your case will be delayed. This results in increased cost and sometimes loss of tactical advantage. Improper service will also delay the obtainment of evidence and delay court assistance such as injunctions etc.
What is a writ?
An official court document, signed by a judge or bearing an official court seal, which commands the person to whom it is addressed, to do something specific. That "person" is typically either a sheriff (who may be instructed to seize property, for example) or a defendant (for whom the writ is the first notice of formal legal action. In these cases, the writ would command the person to answer the charges laid out in the suit, or else judgment may be made against them in their absence).