In a civil lawsuit, the party who files a complaint against a second party is known as the plaintiff. A plaintiff is also called a claimant or a complainant. (The analogue in criminal court is the prosecution.)
By initiating, or filing a lawsuit against a second party, the plaintiff announces his/her intent to seek redress for an injury or loss, for which the party being sued - known as the defendant - is being held responsible. In legal terms, this redress is called a remedy; responsibility on the part of the defendant is more accurately called liability.
The first step a plaintiff must take is to file a formal complaint, or summons. This is a legal document that outlines the nature of the allegations (the wrongdoing on the part of the defendant, which has yet to be proved) as well as a demand for relief - generally a monetary award.
The defendant must then be notified; this notification is usually made by personal delivery of the complaint to the defendant by an officer of the court known as a process server.
Lawsuits do not always involve a full trial, nor even a jury in most cases. Usually, the suit is heard before a judge in a proceeding in which the plaintiff is known as the petitioner and the defendant is called the respondent.
Civil suits can also involve issues that are not adversarial, i.e. one side against another. In such a case, a petitioner may be seeking statutory relief in a matter that requires judicial approval; for example, a homeowner may ask for an exception to an outdated building code that prohibits him from modifying his house or putting up an addition.