Academically speaking, Non-Government Organization (NGO) is a very broad, umbrella term that has a technical definition which includes, non-profit corporations, non-legal (meaning not defined in law, as opposed to 'illegal') entities such as religions or informal community efforts, and also - Political Parties, except in (usually single-party) states where the political party is at least de facto, and often de jure, an extension of the government itself. It does not exclude for-profit entities, but conversationally they're not usually meant to be included unless there's a particular pattern of influence involved.
Because political science academia tends to focus on policymaking and service provision, when the term is used conversationally it has the narrower meaning of such organizations which meaningfully contribute to policy, economic, or social outcomes. This is why sometimes for-profit entities are discussed as NGOs - monopoly firms whose behavior has the de facto effect of dictating policy outcomes in their market are NGOs in this sense.
The Catholic Church is an NGO (at least in the United States), it takes direct policy advocacy stances and provides various social services through it's churches, but it is not part of the United States government in any way.
The International Standards Organization (ISO) is an NGO even though it doesn't directly participate in policy discussion, nor does it provide social services, because wherever governments have been silent as to industrial standards, ISO has created a de facto form of policy which carries considerable weight.
A Credit Union is, technically, an NGO - and here the difference between the strict technical definition versus the conversational definition can be seen: the Credit Union doesn't contribute to policy decisions or outcomes, and arguably doesn't contribute social services - and so is generally not intended to be included when the term is invoked, because scholars care most about entities that meaningful influence policy/economic/social outcomes.
Political Parties, therefore, are NGOs. But not all NGOs are political parties.