Is The New York Times a newspaper or magazine?

No, not at all. They are two different companies and two different publications. "Time" magazine is a weekly news magazine that has been published since 1923. It was founded by a conservative (Henry Luce) but these days is known for being moderate and balanced in its coverage of news. The "New York Times Sunday Magazine" is the magazine supplement to the New York Times newspaper; it offers long-form stories and in-depth features, and appears only in the Sunday edition of the Times.