Definition
Ex parte is Latin for 'from one party.'
Overview
In the law, ex parte is used in several contexts.
Legal Ethics
In legal ethics, ex parte refers to improper contact with a party or a judge. Ethical rules typically forbid a lawyer from contacting the judge or the opposing party without the other party's lawyer also being present. A breach of these rules is referred to as improper ex parte contact.
Civil Procedure
In civil procedure, ex parte is used to refer to motions for orders that can be granted without waiting for a response from the other side. Generally, these are orders that are only in place until further hearings can be held, such as a temporary restraining order.
Typically, a court will be hesitant to make an ex parte motion. This is because the Fifth Amendment and the Fourteenth Amendment guarantee a right to due process, and ex parte motions--due to their exclusion of one party--risk violating the excluded party's right to due process.
Further Reading
For more on ex parte, see this University of Richmond Law Review article, this University of Stanford Law Review article, and this American Bar Association article.