The defining feature of an ex parte motion is that no notice is given to people other than the people making the motion. When a motion is ex parte it is subject to heightened ethical standards for lawyers filing it in U.S. practice. Essentially, a person filing an ex parte motion has to disclose to the court facts and legal arguments against granting the motion that would otherwise not have to be disclosed.
is an ex parte action only made when there is an opposition?
No.
Are motions made by a party in a case without an adversary "ex parte"?
Sometimes, but not always. A motion made that will be ruled upon following notice and hearing to interested parties who are not necessarily adverse, is not ex parte.
For example, if a motion to close a probate estate is made following notice to other relatives and to potential creditors who must respond by a hearing date in a probate case, the motion is not ex parte.
Courts where cases must have an adversary (e.g. criminal and U.S.
civil courts) are naturally excluded.
Ex parte motions in civil and criminal cases aren't uncommon, especially early on in these cases.
A motion requesting issuance of a search or arrest warrant in a criminal case is usually ex parte, as are most requests for temporary restraining orders and for leave to allow substituted service of process in newly filed civil cases.