Q. What's the difference between an editorial, an op-ed (opinion piece), and a letter to the editor of a publication/periodical, such as seen in the New York Times?

In certain contexts, they mean the same thing, but there is some nuance!

An editorial is a periodical article that gives the opinions of the editors or publishers. Many US publishers label them as "Opinion."

Opinion pieces are sometimes editorials, but are often written by members of the community.

Letters to the editor are short (rarely over 200 words) and are meant to be published quickly in response to articles or news that publication recently put out. They're typically written by a reader of that publication, a general person.   

Op-eds, sometimes labeled "Opinion" by places like the New York Times, are usually longer, and means "opposite the editorial page." They, too, are usually written by members of the community, not journalists with the newspaper. They are often experts who have a particular insight into a topic. Op-ed authors can either submit their article for publication or their writing is solicited by the publication itself. 

The first Op-Ed page in The New York Times greeted the world on Sept. 21, 1970. It was so named because it appeared opposite the editorial page and not (as many still believe) because it would offer views contrary to the paper’s. Inevitably, it would do that, too, since its founders were putting out a welcome mat for ideas and arguments from many points on the political, social and cultural spectrums from outside the walls of The Times — to stimulate thought and provoke discussion of public problems.

That important mission remains the same. But it’s time to change the name. The reason is simple: In the digital world, in which millions of Times readers absorb the paper’s journalism online, there is no geographical “Op-Ed,” just as there is no geographical “Ed” for Op-Ed to be opposite to. It is a relic of an older age and an older print newspaper design.

So now, at age 50, the designation will be retired. Editorials will still be called editorials, but the articles written by outside writers will be known as “Guest Essays,” a title that will appear prominently above the headline.

From: "Why The New York Times Is Retiring the Term ‘Op-Ed’", 2021

See the links below to more context and definitions.